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The Pressure of Society on Water Quality: A Land Use Impact Study of Lake Ripley in Oakland, Wisconsin

  • Eutrophication of lakes occurs naturally over time, but the eutrophication rate can be accelerated by human activities. Agriculture land use can negatively impact water quality of lakes due to nutrient pollution. This research investigates the impacts of agricultural land use on the water quality of Lake Ripley in Oakland, Wisconsin from 1993 to 2011. This study performs a regression analysis which incorporates four years of National Land Cover Database (NLCD) data, eight spatial categories based on hydrological flow length across topographic surface, and a weighting technique to calculate land use percentages. The results indicate that the combination of agricultural land use and rainfall variables are significantly related to chlorophyll a and total phosphorus concentrations, while these variables do not appear to affect Secchi depth measurements. Due to the near flat topography of the Lake Ripley watershed, agricultural land use within the two spatial regions closest to Lake Ripley and its inlet stream had the largest impact on Lake Ripley's water quality.

    Citation: Kyle Whalley, Wei Luo. The Pressure of Society on Water Quality: A Land Use Impact Study of Lake Ripley in Oakland, Wisconsin[J]. AIMS Geosciences, 2017, 3(1): 14-36. doi: 10.3934/geosci.2017.1.14

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  • Eutrophication of lakes occurs naturally over time, but the eutrophication rate can be accelerated by human activities. Agriculture land use can negatively impact water quality of lakes due to nutrient pollution. This research investigates the impacts of agricultural land use on the water quality of Lake Ripley in Oakland, Wisconsin from 1993 to 2011. This study performs a regression analysis which incorporates four years of National Land Cover Database (NLCD) data, eight spatial categories based on hydrological flow length across topographic surface, and a weighting technique to calculate land use percentages. The results indicate that the combination of agricultural land use and rainfall variables are significantly related to chlorophyll a and total phosphorus concentrations, while these variables do not appear to affect Secchi depth measurements. Due to the near flat topography of the Lake Ripley watershed, agricultural land use within the two spatial regions closest to Lake Ripley and its inlet stream had the largest impact on Lake Ripley's water quality.


    In this paper we establish and prove two new sufficiency theorems for weak and strong minima for an optimal control problem of Lagrange with fixed end-points, nonlinear dynamics, nonlinear isoperimetric inequality and equality constraints and pointwise mixed nonlinear time-state-control inequality and equality restrictions. The proof of the sufficiency theorems is independent of classical methods used to obtain sufficiency in optimal control problems of this type, see for example [32], where the insertion of the original optimal control problem in a Banach space is a fundamental component in order to obtain the corresponding sufficiency theory; [16], where the construction of a bounded solution of a matrix Riccati equation is crucial in this sufficiency approach; or [8,19], where a verification function and a quadratic function satisfying a Hamilton-Jacobi inequality is an indispensable tool in the sufficiency treatments of these theories. Concretely, the sufficiency theorems of this article state that if an admissible process satisfies a first order sufficient condition related with Pontryagin maximum principle, a similar hypothesis of the necessary Legendre-Clebsch condition, the positivity of a quadratic integral on the cone of critical directions, and several conditions of Weierstrass of some functions, where one of them plays a similar role to the Hamiltonian of the problem, then, the previously mentioned admissible process is a strict local minimum. The set of active indices of the corresponding mixed time-state-control inequality constraints must be piecewise constant on the underlying time interval of consideration, the Lagrange multipliers of the inequality constraints must be nonnegative and in fact they have to be zero whenever the associated index of the Lagrange multiplier is inactive. Additionally, the proposed optimal controls need not be continuous on the underlying interval of time but only measurable, see for example [7,8,9,13,14,15,16,17,18,19,21,22,25,26,27,29,32], where the continuity of the proposed optimal control is a crucial assumption in some sufficiency optimal control theories having the same degree of generality as the problems studied in this article. In contrast, in Examples 2.3 and 2.4, we show how two purely measurable optimal controls comprised with the proposed optimal processes satisfy all the hypotheses of Theorems 2.1 and 2.2 becoming in this way strict local minima.

    Additionally, it is worth mentioning that in these new sufficiency theorems for local minima presented in this paper, all the premises that must be satisfied by an admissible process to become an optimal process, are imposed in the hypotheses established in the theorems, in contrast, with other second order necessary and sufficiency theories which depend upon the verifiability of some preliminary assumptions, see for example [2,3,5,6,11,24], where the necessary second order conditions for optimality depend on some previous hypotheses involving the full rankness of a matrix whose nature arises from the linear independence of vectors whose role are the gradients of the active inequality and equality constraints and where further assumptions involving some notions of regularity or normality of a solution are fundamental hypotheses; or [28], where the corresponding sufficiency theory for optimality depends upon the existence of a continuous function dominating the norm of one of the partial derivatives of the dynamic of the problem. Another remarkable feature presented in this theory concerns the fact that our sufficiency treatment not only provides sufficient conditions for strict local minima but they allow measuring the deviation between admissible costs and optimal costs. This deviation involves a functional playing the role of the square of a norm of the Banach space L1, see for example [1,23], where similar estimations of the growth of the objective functional around the optimal control are established.

    On the other hand, it is worth pointing out the existence of some recent optimal control theories which also study optimal control problems with functional inequality or equality restrictions such as the isoperimetric constraints of this paper. Concretely, in [20], necessary optimality conditions for a Mayer optimal control problem involving semilinear unbounded evolution inclusions and inequality and equality Lipschitzian restrictions are obtained by constructing a sequence of discrete approximations and proving that the optimal solutions of discrete approximations converge uniformly to a given optimal process for the primary continuous-time problem. In [12], necessary and sufficient optimality conditions of Mayer optimal control problems involving differential inclusions and functional inequality constraints are presented and the authors study Mayer optimal control problems with higher order differential inclusions and inequality functional constraints. The necessary conditions for optimality obtained in [12], are important generalizations of associated problems for a first order differential inclusions of optimality settings established in [4,10,20]. The sufficiency conditions obtained in [12] include second order discrete inclusions with inequality end-point constraints. The use of convex and nonsmooth analysis plays a crucial role in this related sufficiency treatment. Moreover, one of the fundamental novelties of the work provided in [12] concerns the derivation of sufficient optimality conditions for Mayer optimal control problems having m-th order ordinary differential inclusions with m3.

    The paper is organized as follows. In Section 2, we pose the problem we shall deal with together with some basic definitions, the statement of the main results and two examples illustrating the sufficiency theorems of the article. Section 3 is devoted to state one auxiliary lemma on which the proof of Theorem 2.1, given in the same section, is strongly based. Section 4 is dedicated to state another auxiliary result on which the proof of Theorem 2.2, once again given in the same section, is based.

    Suppose we are given an interval T:=[t0,t1] in R, two fixed points ξ0, ξ1 in Rn, functions L, Lγ (γ=1,,K) mapping T×Rn×Rm to R, two functions f and φ=(φ1,,φs) mapping T×Rn×Rm to Rn and Rs respectively. Let

    A:={(t,x,u)T×Rn×Rmφα(t,x,u)0(αR),φβ(t,x,u)=0(βS)}

    where R:={1,,r} and S:={r+1,,s} (r=0,1,,s). If r=0 then R= and we disregard statements involving φα. Similarly, if r=s then S= and we disregard statements involving φβ.

    Let {Λn} be a sequence of measurable functions and let Λ be a measurable function. We shall say that the sequence of measurable functions {Λn} converges almost uniformly to a function Λ on T, if given ϵ>0, there exists a measurable set ΥϵT with m(Υϵ)<ϵ such that {Λn} converges uniformly to Λ on TΥϵ. We will also denote uniform convergence by ΛnuΛ, almost uniform convergence by ΛnauΛ, strong convergence in Lp by ΛnLpΛ and weak convergence in Lp by ΛnLpΛ. From now on we shall not relabel the subsequences of a given sequence since this fact will not alter our results.

    It will be assumed throughout the paper that L, Lγ (γ=1,,K), f and φ have first and second derivatives with respect to x and u. Also, if we denote by b(t,x,u) either L(t,x,u), Lγ(t,x,u) (γ=1,,K), f(t,x,u), φ(t,x,u) or any of their partial derivatives of order less or equal than two with respect to x and u, we shall assume that if B is any bounded subset of T×Rn×Rm, then |b(B)| is a bounded subset of R. Additionally, we shall assume that if {(Φq,Ψq)} is any sequence in AC(T;Rn)×L(T;Rm) such that for some ΥT measurable and some (Φ0,Ψ0)AC(T;Rn)×L(T;Rm), (Φq,Ψq)L(Φ0,Ψ0) on Υ, then for all qN, b(,Φq(),Ψq()) is measurable on Υ and

    b(,Φq(),Ψq())Lb(,Φ0(),Ψ0()) on Υ.

    Note that all conditions given above are satisfied if the functions L, Lγ (γ=1,,K), f and φ and their first and second derivatives with respect to x and u are continuous on T×Rn×Rm.

    The fixed end-point optimal control problem we shall deal with, denoted by (P), is that of minimizing the functional

    I(x,u):=t1t0L(t,x(t),u(t))dt

    over all couples (x,u) with x:TRn absolutely continuous and u:TRm essentially bounded, satisfying the constraints

    {˙x(t)=f(t,x(t),u(t))(a.e. in T).x(t0)=ξ0,x(t1)=ξ1.Ii(x,u):=t1t0Li(t,x(t),u(t))dt0(i=1,,k).Ij(x,u):=t1t0Lj(t,x(t),u(t))dt=0(j=k+1,,K).(t,x(t),u(t))A(tT).

    Denote by X the space of all absolutely continuous functions mapping T to Rn and by Uc:=L(T;Rc) (cN). Elements of X×Um will be called processes and a process (x,u) is admissible if it satisfies the constraints. A process (x,u) solves (P) if it is admissible and I(x,u)I(y,v) for all admissible processes (y,v). An admissible process (x,u) is called a strong minimum of (P) if it is a minimum of I with respect to the norm

    x:=suptT|x(t)|,

    that is, if for any ϵ>0, I(x,u)I(y,v) for all admissible processes (y,v) satisfying yx<ϵ. An admissible process (x,u) is called a weak minimum of (P) if it is a minimum of I with respect to the norm

    (x,u):=x+u,

    that is, if for any ϵ>0, I(x,u)I(y,v) for all admissible processes (y,v) satisfying (y,v)(x,u)<ϵ. It is a strict minimum if I(x,u)=I(y,v) only in case (x,u)=(y,v). Note that the crucial difference between strong and weak minima is that in the former, if I affords a strong minimum at (x0,u0), then, if (x,u) is admissible and it is sufficiently close to (x0,u0), in the sense that the quantity xx0 is sufficiently small, then I(x,u)I(x0,u0), meanwhile for the latter, if (x,u) is admissible and it is sufficiently close to (x0,u0), in the sense that the quantities xx0, uu0 are sufficiently small, then I(x,u)I(x0,u0).

    The following definitions will be useful in order to continue with the development of this theory.

    For any (x,u)X×Um we shall use the notation (˜x(t)) to represent (t,x(t),u(t)). Similarly (˜x0(t)) represents (t,x0(t),u0(t)). Throughout the paper the notation "" will denote transpose.

    Given K real numbers λ1,,λK, consider the functional I0:X×UmR defined by

    I0(x,u):=I(x,u)+Kγ=1λγIγ(x,u)=t1t0L0(˜x(t))dt,

    where L0:T×Rn×RmR is given by

    L0(t,x,u):=L(t,x,u)+Kγ=1λγLγ(t,x,u).

    For all (t,x,u,ρ,μ)T×Rn×Rm×Rn×Rs, set

    H(t,x,u,ρ,μ):=ρf(t,x,u)L0(t,x,u)μφ(t,x,u).

    Given ρX and μUs define, for all (t,x,u)T×Rn×Rm,

    F0(t,x,u):=H(t,x,u,ρ(t),μ(t))˙ρ(t)x

    and let

    J0(x,u):=ρ(t1)ξ1ρ(t0)ξ0+t1t0F0(˜x(t))dt.

    Consider the first variations of J0 and Iγ (γ=1,,K) with respect to (x,u)X×Um over (y,v)X×L2(T;Rm) which are given, respectively, by

    J0((x,u);(y,v)):=t1t0{F0x(˜x(t))y(t)+F0u(˜x(t))v(t)}dt,
    Iγ((x,u);(y,v)):=t1t0{Lγx(˜x(t))y(t)+Lγu(˜x(t))v(t)}dt.

    The second variation of J0 with respect to (x,u)X×Um over (y,v)X×L2(T;Rm) is given by

    J0((x,u);(y,v)):=t1t02Ω0(˜x(t);t,y(t),v(t))dt

    where, for all (t,y,v)T×Rn×Rm,

    2Ω0(˜x(t);t,y,v):=yF0xx(˜x(t))y+2yF0xu(˜x(t))v+vF0uu(˜x(t))v.

    Denote by E0 the Weierstrass excess function of F0, given by

    E0(t,x,u,v):=F0(t,x,v)F0(t,x,u)F0u(t,x,u)(vu).

    Similarly, the Weierstrass excess function of Lγ (γ=1,,K) corresponds to

    Eγ(t,x,u,v):=Lγ(t,x,v)Lγ(t,x,u)Lγu(t,x,u)(vu).

    For all (x,u)X×L1(T;Rm) let

    D(x,u):=max{D1(x),D2(u)}

    where

    D1(x):=V(x(t0))+t1t0V(˙x(t))dtandD2(u):=t1t0V(u(t))dt,

    where V(π):=(1+|π|2)1/21 with π:=(π1,,πn)Rn or π:=(π1,,πm)Rm.

    Finally, for all (t,x,u)T×Rn×Rm, denote by

    Ia(t,x,u):={αRφα(t,x,u)=0},

    the set of active indices of (t,x,u) with respect to the mixed inequality constraints. For all (x,u)X×Um, denote by

    ia(x,u):={i=1,,kIi(x,u)=0},

    the set of active indices of (x,u) with respect to the isoperimetric inequality constraints. Given (x,u)X×Um, let Y(x,u) be the set of all (y,v)X×L2(T;Rm) satisfying

    {˙y(t)=fx(˜x(t))y(t)+fu(˜x(t))v(t)(a.e. in T),y(ti)=0(i=0,1).Ii((x,u);(y,v))0(iia(x,u)),Ij((x,u);(y,v))=0(j=k+1,,K).φαx(˜x(t))y(t)+φαu(˜x(t))v(t)0(a.e. in T,αIa(˜x(t))).φβx(˜x(t))y(t)+φβu(˜x(t))v(t)=0(a.e. in T,βS).

    The set Y(x,u) is called the cone of critical directions along (x,u).

    Now we are in a position to state the main results of the article, two sufficiency results for strict local minima of problem (P). Given an admissible process (x0,u0) where the proposed optimal controls u0 need not be continuous but only measurable, the hypotheses include, two conditions related with Pontryagin maximum principle, a similar assumption of the necessary Legendre-Clebsch condition, the positivity of the second variation on the cone of critical directions and some conditions involving the Weierstrass functions delimiting problem (P). It is worth observing that the sufficiency theorems not only give sufficient conditions for strict local minima but also provides some information concerning the deviation between optimal and feasible costs. In the measure of this deviation are involved the functionals Di (i=1,2) which play the role of the square of the norm of the Banach space L1.

    The following theorem provides sufficient conditions for a strict strong minimum of problem (P).

    Theorem 2.1 Let (x0,u0) be an admissible process. Assume that Ia(˜x0()) is piecewise constant on T, suppose that there exist ρX, μUs with μα(t)0 and μα(t)φα(˜x0(t))=0 (αR,tT), two positive numbers δ,ϵ, and multipliers λ1,,λK with λi0 and λiIi(x0,u0)=0 (i=1,,k) such that

    ˙ρ(t)=Hx(˜x0(t),ρ(t),μ(t)) (a.e.in T),
    Hu(˜x0(t),ρ(t),μ(t))=0 (tT),

    and the following holds:

    (ⅰ) Huu(˜x0(t),ρ(t),μ(t))0 (a.e.in T).

    (ⅱ) J0((x0,u0);(y,v))>0 for all (y,v)(0,0), (y,v)Y(x0,u0).

    (ⅲ) If (x,u) is admissible with xx0<ϵ, then

    a. E0(t,x(t),u0(t),u(t))0 (a.e.in T).

    b. t1t0E0(t,x(t),u0(t),u(t))dtδmax{t1t0V(˙x(t)˙x0(t))dt,t1t0V(u(t)u0(t))dt}.

    c. t1t0E0(t,x(t),u0(t),u(t))dtδ|t1t0Eγ(t,x(t),u0(t),u(t))dt| (γ=1,,K).

    In this case, there exist θ1,θ2>0 such that if (x,u) is admissible with xx0<θ1,

    I(x,u)I(x0,u0)+θ2D(xx0,uu0).

    In particular, (x0,u0) is a strict strong minimum of (P).

    The theorem below gives sufficient conditions for weak minima of problem (P).

    Theorem 2.2 Let (x0,u0) be an admissible process. Assume that Ia(˜x0()) is piecewise constant on T, suppose that there exist ρX, μUs with μα(t)0 and μα(t)φα(˜x0(t))=0 (αR,tT), two positive numbers δ,ϵ, and multipliers λ1,,λK with λi0 and λiIi(x0,u0)=0 (i=1,,k) such that

    ˙ρ(t)=Hx(˜x0(t),ρ(t),μ(t)) (a.e.in T),
    Hu(˜x0(t),ρ(t),μ(t))=0 (tT),

    and the following holds:

    (ⅰ) Huu(˜x0(t),ρ(t),μ(t))0 (a.e.in T).

    (ⅱ) J0((x0,u0);(y,v))>0 for all (y,v)(0,0), (y,v)Y(x0,u0).

    (ⅲ) If (x,u) is admissible with (x,u)(x0,u0)<ϵ, then

    a. t1t0E0(t,x(t),u0(t),u(t))dtδt1t0V(u(t)u0(t))dt.

    b. t1t0E0(t,x(t),u0(t),u(t))dtδ|t1t0Eγ(t,x(t),u0(t),u(t))dt| (γ=1,,K).

    In this case, there exist θ1,θ2>0 such that if (x,u) is admissible with (x,u)(x0,u0)<θ1,

    I(x,u)I(x0,u0)+θ2D2(uu0).

    In particular, (x0,u0) is a strict weak minimum of (P).

    Examples 2.3 and 2.4 illustrate Theorems 2.1 and 2.2 respectively. It is worth mentioning that the sufficiency theory of [28] cannot be applied in both examples. Indeed, if f denotes the dynamic of the problems, as one readily verifies, in both examples, we have that

    fu(t,x,u)=(u2,u1) for all (t,x,u)[0,1]×R×R2,

    and hence, it does not exist a continuous function ψ:[0,1]×RR such that

    |fu(t,x,u)|ψ(t,x) for all (t,x,u)[0,1]×R×R2.

    Example 2.3 Let u02:[0,1]R be any measurable function whose codomain belongs to the set {1,1}.

    Consider problem (P) of minimizing

    I(x,u):=10{sinh(u1(t))+u21(t)cos(2πu2(t))x2(t)}dt

    over all couples (x,u) with x:[0,1]R absolutely continuous and u:[0,1]R2 essentially bounded satisfying the constraints

    {x(0)=x(1)=0.˙x(t)=u1(t)u2(t)+12x(t) (a.e.in[0,1]).I1(x,u):=10{14x2(t)+x(t)u1(t)u2(t)}dt0.(t,x(t),u(t))A (t[0,1])

    where

    A:={(t,x,u)[0,1]×R×R2u10, (u2u02(t))21, u22=1}.

    For this case, T=[0,1], n=1, m=2, r=2, s=3, k=K=1, ξ0=ξ1=0,

    L(t,x,u)=sinh(u1)+u21cos(2πu2)x2,f(t,x,u)=u1u2+12x,
    L1(t,x,u)=14x2+xu1u2,L0(t,x,u)=sinh(u1)+u21cos(2πu2)x2+λ1[14x2+xu1u2],
    φ1(t,x,u)=u1,φ2(t,x,u)=(u2u02(t))21,φ3(t,x,u)=u221.

    Clearly, L, L1, f and φ=(φ1,φ2,φ3) satisfy the hypotheses imposed in the statement of the problem.

    Also, as one readily verifies, the process (x0,u0)=(x0,u01,u02)(0,0,u02) is admissible.

    Moreover,

    H(t,x,u,ρ,μ)=ρu1u2+12ρxsinh(u1)u21cos(2πu2)+x2λ1[14x2+xu1u2]+μ1u1μ2[(u2u02(t))21]μ3[u221],
    Hx(t,x,u,ρ,μ)=12ρ+2xλ1[12x+u1u2],
    Hu(t,x,u,ρ,μ)=(ρu2cosh(u1)2u1cos(2πu2)λ1xu2+μ1ρu1+2πu21sin(2πu2)λ1xu12μ2(u2u02(t))2μ3u2).

    Therefore, if we set ρ0, μ11, μ2=μ30 and λ1=0, we have

    ˙ρ(t)=Hx(˜x0(t),ρ(t),μ(t)) (a.e. in T),Hu(˜x0(t),ρ(t),μ(t))=(0,0) (tT),

    and hence the first order sufficient conditions involving the Hamiltonian of problem (P) are verified. Moreover, if we set R:={1,2}, observe that

    λ10,λ1I1(x0,u0)=0,
    μα(t)0,μα(t)φα(˜x0(t))=0(αR,tT).

    Additionally, Ia(˜x0()){1} is constant on T. Also, it is readily seen that for all tT,

    Huu(˜x0(t),ρ(t),μ(t))=(2000),

    and so condition (ⅰ) of Theorem 2.1 is satisfied. Observe that, for all tT,

    fx(˜x0(t))=12,fu(˜x0(t))=(u02(t),0),L1x(˜x0(t))=0,L1u(˜x0(t))=(0,0),
    φ1x(˜x0(t))=0,φ1u(˜x0(t))=(1,0),φ3x(˜x0(t))=0,φ3u(˜x0(t))=(0,2u02(t)).

    Thus, Y(x0,u0) is given by all (y,v)X×L2(T;R2) satisfying

    {y(0)=y(1)=0.˙y(t)=12y(t)+u02(t)v1(t) (a.e. in T).v1(t)0 (a.e. in T).2u02(t)v2(t)=0 (a.e. in T).

    Moreover, note that, for all (t,x,u)T×R×R2,

    F0(t,x,u)=H(t,x,u,ρ(t),μ(t))˙ρ(t)x=sinh(u1)+u21cos(2πu2)x2u1,

    and so, for all tT,

    F0xx(˜x0(t))=2,F0xu(˜x0(t))=(0,0),F0uu(˜x0(t))=(2000).

    Consequently, we have

    12J0((x0,u0);(y,v))=10{v21(t)y2(t)}dt=10{(˙y(t)12y(t))2y2(t)}dt=10{˙y2(t)y(t)˙y(t)34y2(t)}dt=10{˙y2(t)34y2(t)}dt>0

    for all (y,v)(0,0), (y,v)Y(x0,u0). Hence, condition (ⅱ) of Theorem 2.1 is verified.

    Additionally, observe that for all (x,u) admissible and all tT,

    E0(t,x(t),u0(t),u(t))=sinh(u1(t))+u21(t)cos(2πu2(t))u1(t)u21(t)cos(2πu02(t))=u21(t)0,

    and then, condition (ⅲ)(a) of Theorem 2.1 is satisfied for any ϵ>0. Now, if (x,u) is admissible, then

    uu0=(u1u01,u2u02)=(u1,u02u02)=(u1,0)

    and so, if (x,u) is admissible,

    10E0(t,x(t),u0(t),u(t))dt10u21(t)dt10V(u1(t))dt=10V(u(t)u0(t))dt.

    Also, if (x,u) is admissible, then

    10E0(t,x(t),u0(t),u(t))dt10u21(t)dt=10{(u1(t)u2(t)+12x(t))2x(t)u1(t)u2(t)14x2(t)}dt=10{˙x2(t)x(t)u1(t)u2(t)14x2(t)}dt10˙x2(t)dt10V(˙x(t)˙x0(t))dt.

    Therefore, if (x,u) is admissible, then

    10E0(t,x(t),u0(t),u(t))dtmax{10V(˙x(t)˙x0(t))dt,10V(u(t)u0(t))dt},

    and hence, condition (ⅲ)(b) of Theorem 2.1 is verified for any ϵ>0 and δ=1. Finally, if (x,u) is admissible, note that

    |10E1(t,x(t),u0(t),u(t))dt|=|10x(t)u1(t)u2(t)dt|=|10{x(t)˙x(t)12x2(t)}dt|=|1012x2(t)dt|=1210x2(t)dt10˙x2(t)dt=10(u1(t)u2(t)+12x(t))2dt=10u21(t)dt+10{x(t)u1(t)u2(t)+14x2(t)}dt10u21(t)dt+10x(t)˙x(t)dt=10u21(t)dt10E0(t,x(t),u0(t),u(t))dt,

    implying that condition (ⅲ)(c) of Theorem 2.1 holds for any ϵ>0 and δ=1. By Theorem 2.1, (x0,u0) is a strict strong minimum of (P).

    Example 2.4 Consider problem (P) of minimizing

    I(x,u):=10{sinh(u1(t)+u1(t)x3(t))+12u21(t)cos(2πu2(t))cosh(x(t))+1}dt

    over all couples (x,u) with x:[0,1]R absolutely continuous and u:[0,1]R2 essentially bounded satisfying the constraints

    {x(0)=x(1)=0.˙x(t)=u1(t)u2(t)+x(t) (a.e.in[0,1]).I1(x,u):=10{sin(u1(t))sinh(u1(t)+u1(t)x3(t))}dt0.(t,x(t),u(t))A (t[0,1])

    where

    A:={(t,x,u)[0,1]×R×R2sin(u1)0, u22=1}.

    For this case, T=[0,1], n=1, m=2, r=1, s=2, k=K=1, ξ0=ξ1=0,

    L(t,x,u)=sinh(u1+u1x3)+12u21cos(2πu2)cosh(x)+1,f(t,x,u)=u1u2+x,
    L_1(t, x, u) = \sin(u_1)-\sinh(u_1+u_1x^3),
    L_0(t, x, u) = \sinh(u_1+u_1x^3)+{{ {{1} \over {2}}}} u_1^2\cos(2\pi u_2)-\cosh(x)+1+\lambda_1[\sin(u_1)-\sinh(u_1+u_1x^3)],
    \varphi_1(t, x, u) = -\sin(u_1), \quad \varphi_2(t, x, u) = u_2^2-1.

    Clearly, L , L_1 , f and \varphi = (\varphi_1, \varphi_2) satisfy the hypotheses imposed in the statement of the problem.

    Let u_{02}\colon T\to{{\bf R}} be any measurable function whose codomain belongs to the set \{-1, 1\}.

    Clearly, the process (x_0, u_0) = (x_0, u_{01}, u_{02})\equiv(0, 0, u_{02}) is admissible.

    Moreover,

    \begin{eqnarray*} H(t, x, u, \rho, \mu) & = & \rho u_1u_2 + \rho x -\sinh(u_1+u_1x^3)-{{ {{1} \over {2}}}} u_1^2\cos(2\pi u_2)+\cosh(x)-1 \\ && -\lambda_1[\sin(u_1)-\sinh(u_1+u_1x^3)] +\mu_1\sin(u_1)-\mu_2[u_2^2-1], \end{eqnarray*}
    H_x(t, x, u, \rho, \mu) = \rho-3x^2u_1\cosh(u_1+u_1x^3)+\sinh(x)+3\lambda_1x^2u_1\cosh(u_1+u_1x^3),
    \begin{eqnarray*} H_{u_1}(t, x, u, \rho, \mu) & = & \rho u_2-[1+x^3]\cosh(u_1+u_1x^3)-u_1\cos(2\pi u_2) \\ && -\lambda_1[\cos(u_1)-\{1+x^3\}\cosh(u_1+u_1x^3)]+\mu_1\cos(u_1), \end{eqnarray*}
    H_{u_2}(t, x, u, \rho, \mu) = \rho u_1+\pi u_1^2\sin(2\pi u_2)-2\mu_2u_2.

    Therefore, if we set \rho\equiv0 , \mu_1\equiv1 , \mu_2\equiv0 and \lambda_1 = 0 , we have

    \dot\rho(t) = -H_x({{\tilde x}}_0(t), \rho(t), \mu(t)) \ ({\hbox{a.e. in}\ T}), \quad H_u({{\tilde x}}_0(t), \rho(t), \mu(t)) = (0, 0) \ (t\in T),

    and hence the first order sufficient conditions involving the Hamiltonian of problem (P) are verified. Additionally, observe that

    \lambda_1\ge0, \quad \lambda_1I_1(x_0, u_0) = 0,
    \mu_1(t)\ge0, \quad \mu_1(t)\varphi_1({{\tilde x}}_0(t)) = 0 \quad (t\in T).

    Also, {{\cal I}}_a({{\tilde x}}_0(\cdot))\equiv\{1\} is constant on T . Moreover, it is readily seen that for all t\in T ,

    H_{uu}({{\tilde x}}_0(t), \rho(t), \mu(t)) = \left(\begin{array}{cc} -1 & 0 \\ 0 & 0 \\ \end{array}\right),

    and so condition (ⅰ) of Theorem 2.2 is satisfied. Observe that, for all t\in T ,

    f_x({{\tilde x}}_0(t)) = 1, \quad f_u({{\tilde x}}_0(t)) = (u_{02}(t), 0), \quad L_{1x}({{\tilde x}}_0(t)) = 0, \quad L_{1u}({{\tilde x}}_0(t)) = (0, 0),
    \varphi_{1x}({{\tilde x}}_0(t)) = 0, \quad \varphi_{1u}({{\tilde x}}_0(t)) = (-1, 0), \quad \varphi_{2x}({{\tilde x}}_0(t)) = 0, \quad \varphi_{2u}({{\tilde x}}_0(t)) = (0, 2u_{02}(t)).

    Thus, {{\cal Y}}(x_0, u_0) is given by all (y, v)\in{{\cal X}}\times L^2(T; {{\bf R}}^2) satisfying

    \left\{\begin{array}{l} y(0) = y(1) = 0. \\ {{\dot y}}(t) = y(t)+u_{02}(t)v_1(t) \ ({\hbox{a.e. in}\ T}). \\ -v_1(t)\le0 \ ({\hbox{a.e. in}\ T}). \\ 2u_{02}(t)v_2(t) = 0 \ ({\hbox{a.e. in}\ T}). \\ \end{array} \right.

    Also, note that, for all (t, x, u)\in T\times{{\bf R}}\times{{\bf R}}^2 ,

    F_0(t, x, u) = -H(t, x, u, \rho(t), \mu(t))-\dot\rho(t)x = \sinh(u_1+u_1x^3)+{{ {{1} \over {2}}}} u_1^2\cos(2\pi u_2)-\cosh(x)+1-\sin(u_1),

    and so, for all t\in T ,

    F_{0xx}({{\tilde x}}_0(t)) = -1, \quad F_{0xu}({{\tilde x}}_0(t)) = (0, 0), \quad F_{0uu}({{\tilde x}}_0(t)) = \left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \\ \end{array}\right).

    Consequently, we have

    \begin{eqnarray*} {J^{\prime\prime}}_0((x_0, u_0);(y, v)) & = & {\int_0^1} \{ v_1^2(t)-y^2(t) \} dt = {\int_0^1} \{ ({{\dot y}}(t)-y(t))^2-y^2(t) \} dt \\ & = & {\int_0^1} \{ {{\dot y}}^2(t)-2y(t){{\dot y}}(t) \} dt = {\int_0^1} {{\dot y}}^2(t) dt > 0 \end{eqnarray*}

    for all (y, v)\not = (0, 0) , (y, v)\in{{\cal Y}}(x_0, u_0) . Hence, condition (ⅱ) of Theorem 2.2 is verified.

    Additionally, observe that for any \epsilon\in(0, 1) , all (x, u) admissible satisfying \|(x, u)-(x_0, u_0)\| < \epsilon and all t\in T ,

    \begin{eqnarray*} E_0(t, x(t), u_0(t), u(t)) & = & \sinh(u_1(t)+u_1(t)x^3(t))+{{ {{1} \over {2}}}} u_1^2(t)\cos(2\pi u_2(t))-\sin(u_1(t)) \\ & = & \sinh(u_1(t)+u_1(t)x^3(t))+{{ {{1} \over {2}}}} u_1^2(t)\cos(2\pi u_{02}(t))-\sin(u_1(t)) \\ & = & \sinh(u_1(t)+u_1(t)x^3(t))+{{ {{1} \over {2}}}} u_1^2(t)-\sin(u_1(t)). \\ \end{eqnarray*}

    Therefore, for any \epsilon\in(0, 1) and all (x, u) admissible satisfying \|(x, u)-(x_0, u_0)\| < \epsilon ,

    \begin{eqnarray*} && {\int_0^1} E_0(t, x(t), u_0(t), u(t)) dt = {\int_0^1} \{ \sinh(u_1(t)+u_1(t)x^3(t))-\sin(u_1(t))+{{ {{1} \over {2}}}} u_1^2(t) \} dt \\ &\ge& {\int_0^1} {{ {{1} \over {2}}}} u_1^2(t)dt \ge {\int_0^1} V(u_1(t))dt = {\int_0^1} V(u(t)-u_0(t))dt, \end{eqnarray*}

    and hence condition (ⅲ)(a ^\prime ) of Theorem 2.2 is satisfied for any \epsilon\in(0, 1) and \delta = 1 .

    Finally, if (x, u) is admissible, note that

    {\int_0^1} E_0(t, x(t), u_0(t), u(t))dt \ge \biggl| {\int_0^1} \{ \sinh(u_1(t)+u_1(t)x^3(t))-\sin(u_1(t)) \}dt \biggr| = \biggl| {\int_0^1} E_1(t, x(t), u_0(t), u(t))dt \biggr|,

    implying that condition (ⅲ)(b ^\prime ) of Theorem 2.2 holds for any {\epsilon > 0} and \delta = 1 . By Theorem 2.2, (x_0, u_0) is a strict weak minimum of (P).

    In this section we shall prove Theorem 2.1. We first state an auxiliary result whose proof is given in Lemmas 2–4 of [31].

    In the following lemma we shall assume that we are given z_0: = (x_0, u_0)\in {{\cal X}}\times L^1(T; {{\bf R}^m}) and a subsequence \{z_q: = (x_q, u_q)\} in {{\cal X}}\times L^1(T; {{\bf R}^m}) such that

    \lim\limits_{q\to\infty}D(z_q-z_0) = 0 \quad\hbox{and}\quad d_q: = [2D(z_q-z_0)]^{1/2} > 0\quad(q\in{{\bf N}}).

    For all q\in{{\bf N}} , set

    y_q: = {{{x_q-x_0}} \over {{d_q}}} \quad\hbox{and}\quad v_q: = {{{u_q-u_0}} \over {{d_q}}}.

    For all q\in{{\bf N}} , define

    W_q: = \max\{W_{1q}, W_{2q}\}

    where

    W_{1q}: = [1+{{ {{1} \over {2}}}} V({{\dot x}}_q-{{\dot x}}_0)]^{1/2} \quad\hbox{and}\quad W_{2q}: = [1+{{ {{1} \over {2}}}} V(u_q-u_0)]^{1/2}.

    As we mentioned in the introduction, we do not relabel the subsequences of a given sequence since as one readily verifies this fact will not alter our results.

    Lemma 3.1

    \quad {\rm\bf a.} For some v_0\in L^2(T; {{\bf R}^m}) and some subsequence of \{z_q\} , v_q \buildrel L^1 \over \rightharpoonup v_0 on T . Even more, u_q \buildrel \hbox{au} \over \longrightarrow u_0 on T .

    \quad {\rm\bf b.} There exist \zeta_0\in L^2(T; {{\bf R}^n}) , \bar y_0\in{{\bf R}^n} , and some subsequence of \{z_q\} , such that {{\dot y}}_q \buildrel L^1 \over \rightharpoonup \zeta_0 on T . Moreover, if y_0(t): = \bar y_0+\int_{t_0}^t\zeta_0(\tau)d\tau (t\in T) , then y_q \buildrel \hbox{u} \over \longrightarrow y_0 on T .

    \quad {\rm\bf c.} Let \Upsilon\subset T be measurable and suppose that W_q \buildrel \hbox{u} \over \longrightarrow 1 on \Upsilon . Let R_q, R_0\in L^\infty(\Upsilon; {{\bf R}}^{m\times m}) , assume that R_q \buildrel \hbox{u} \over \longrightarrow R_0 on \Upsilon , R_0(t)\ge0 (t\in \Upsilon) , and let v_0 be the function considered in condition (a) of Lemma 3.1. Then,

    \liminf\limits_{q\to\infty}\int_\Upsilon v_q^\ast(t)R_q(t)v_q(t) dt \ge \int_\Upsilon v_0^\ast(t)R_0(t)v_0(t) dt.

    Proof. The proof of Theorem 2.1 will be made by contraposition, that is, we shall assume that for all \theta_1, \theta_2 > 0 , there exists an admissible process (x, u) such that

    \begin{align} \|x-x_0\| < \theta_1 \quad\hbox{and} \quad I(x, u) < I(x_0, u_0)+\theta_2 D(x-x_0, u-u_0). \end{align} (1)

    Also, we are going to assume that all the hypotheses of Theorem 2.1 are satisfied with the exception of hypothesis (ⅱ) and we will obtain the negation of condition (ⅱ) of Theorem 2.1. First of all, note that since

    \mu_\alpha(t)\ge0 \ (\alpha\in R, t\in T) \quad\hbox{and}\quad \lambda_i\ge0 \ (i = 1, \ldots, k),

    if (x, u) is admissible, then I(x, u)\ge J_0(x, u) . Also, since

    \mu_\alpha(t)\varphi_\alpha({{\tilde x}}_0(t)) = 0 \ (\alpha\in R, t\in T) \quad\hbox{and}\quad \lambda_i I_i(x_0, u_0) = 0 \ (i = 1, \ldots, k),

    then I(x_0, u_0) = J_0(x_0, u_0) . Thus, (1) implies that for all \theta_1, \theta_2 > 0 , there exists (x, u) admissible with \|x-x_0\| < \theta_1 and

    \begin{align} J_0(x, u) < J_0(x_0, u_0)+\theta_2 D(x-x_0, u-u_0). \end{align} (2)

    Let z_0: = (x_0, u_0) . Note that, for all admissible processes z = (x, u) ,

    \begin{align} J_0(z) = J_0(z_0)+{J^\prime}_0(z_0;z-z_0)+{{\cal K}}_0(z)+{{\cal E}}_0(z) \end{align} (3)

    where

    {{\cal E}}_0(x, u): = {\int_{t_0}^{t_1}} E_0(t, x(t), u_0(t), u(t))dt,
    {{\cal K}}_0(x, u): = {\int_{t_0}^{t_1}} \{ M_0(t, x(t)) + [u^\ast(t)-u_0^\ast(t)] N_0(t, x(t)) \}dt,

    and the functions M_0 and N_0 are given by

    M_0(t, y): = F_0(t, y, u_0(t)) - F_0({{\tilde x}}_0(t)) - F_{0x}({{\tilde x}}_0(t))(y-x_0(t)),
    N_0(t, y): = F_{0u}^\ast(t, y, u_0(t)) - F_{0u}^\ast({{\tilde x}}_0(t)).

    We have,

    M_0(t, y) = {{ {{1} \over {2}}}}[y^\ast-x_0^\ast(t)] P_0(t, y)(y-x_0(t)), \quad N_0(t, y) = Q_0(t, y)(y-x_0(t)),

    where

    P_0(t, y): = 2\int_0^1(1-\lambda)F_{0xx}(t, x_0(t)+\lambda[y-x_0(t)], u_0(t))d\lambda,
    Q_0(t, y): = \int_0^1F_{0ux}(t, x_0(t)+\lambda[y-x_0(t)], u_0(t))d\lambda.

    Now, as in [28], choose \nu > 0 such that for all z = (x, u) admissible with \|x-x_0\| < 1 ,

    \begin{align} |{{\cal K}}_0(x, u)|\le\nu\|x-x_0\|[1+D(z-z_0)]. \end{align} (4)

    Now, by (2), for all q\in{{\bf N}} there exists z_q: = (x_q, u_q) admissible such that

    \begin{align} \|x_q-x_0\| < \epsilon, \quad \|x_q-x_0\| < {{{1}} \over {{q}}}, \quad J_0(z_q)-J_0(z_0) < {{{1}} \over {{q}}}D(z_q-z_0). \end{align} (5)

    The last inequality of (5) implies that z_q\not = z_0 and so for all q\in{{\bf N}} ,

    d_q: = [2D(z_q-z_0)]^{1/2} > 0.

    Since

    \dot\rho(t) = -H_x^\ast({{\tilde x}}_0(t), \rho(t), \mu(t)) \ ({\hbox{a.e. in}\ T}), \quad H_u^\ast({{\tilde x}}_0(t), \rho(t), \mu(t)) = 0 \ (t\in T),

    it follows that {J^\prime}_0(z_0;(y, v)) = 0 for all (y, v)\in{{\cal X}}\times L^2(T; {{\bf R}^m}) . With this in mind, by (3), condition (ⅲ)(b) of Theorem 2.1, (4) and (5),

    J_0(z_q)-J_0(z_0) = {{\cal K}}_0(z_q)+{{\cal E}}_0(z_q) \ge -\nu\|x_q-x_0\|+D(z_q-z_0)(\delta-\nu\|x_q-x_0\|).

    By (5), for all q\in{{\bf N}} ,

    D(z_q-z_0)\biggl( \delta - {{{1}} \over {{q}}} - {{{\nu}} \over {{q}}} \biggr) < {{{\nu}} \over {{q}}}

    and hence

    \lim\limits_{q\to\infty}D(z_q-z_0) = 0.

    For all q\in{{\bf N}} , define

    y_q: = {{{x_q-x_0}} \over {{d_q}}}\quad\hbox{and}\quad v_q: = {{{u_q-u_0}} \over {{d_q}}}.

    By condition (a) of Lemma 3.1, there exist v_0\in L^2(T; {{\bf R}^m}) and a subsequence of \{z_q\} such that v_q \buildrel L^1 \over \rightharpoonup v_0 on T . By condition (b) of Lemma 3.1, there exist \zeta_0\in L^2(T; {{\bf R}^n}) , \bar y_0\in{{\bf R}^n} and a subsequence of \{z_q\} such that, if for all t\in T , y_0(t): = \bar y_0+\int_{t_0}^t\zeta_0(\tau)d\tau , then y_q \buildrel \hbox{u} \over \longrightarrow y_0 on T .

    We claim that

    ⅰ. {J^{\prime\prime}}_0(z_0;(y_0, v_0))\le0 , (y_0, v_0)\not = (0, 0) .

    ⅱ. {{\dot y}}_0(t) = f_x({{\tilde x}}_0(t))y_0(t)+f_u({{\tilde x}}_0(t))v_0(t) ({\hbox{a.e. in}\ T}) , y_0(t_i) = 0 (i = 0, 1) .

    ⅲ. {I^\prime}_i(z_0;(y_0, v_0))\le0 (i\in i_a(z_0)) , {I^\prime}_j(z_0;(y_0, v_0)) = 0 (j = k+1, \ldots, K) .

    ⅳ. \varphi_{\alpha x}({{\tilde x}}_0(t))y_0(t)+\varphi_{\alpha u}({{\tilde x}}_0(t))v_0(t)\le0 ({\hbox{a.e. in}\ T}, \, \alpha\in{{\cal I}}_a({{\tilde x}}_0(t))) .

    ⅴ. \varphi_{\beta x}({{\tilde x}}_0(t))y_0(t)+\varphi_{\beta u}({{\tilde x}}_0(t))v_0(t) = 0 ({\hbox{a.e. in}\ T}, \, \beta\in S) .

    Indeed, the equalities y_0(t_i) = 0 (i = 0, 1) follow from the definition of y_q , the admissibility of z_q and the fact that y_q \buildrel \hbox{u} \over \longrightarrow y_0 on T .

    For all q\in{{\bf N}} , we have

    {{{{{\cal K}}_0(z_q)}} \over {{d_q^2}}} = {\int_{t_0}^{t_1}}\biggl \{ {{{M_0(t, x_q(t))}} \over {{d_q^2}}} + v_q^\ast(t) {{{N_0(t, x_q(t))}} \over {{d_q}}} \biggr \}dt.

    By condition (b) of Lemma 3.1,

    {{{M_0(\cdot, x_q(\cdot))}} \over {{d_q^2}}} \buildrel L^\infty \over \longrightarrow {{ {{1} \over {2}}}} y_0^\ast(\cdot) F_{0xx}({{\tilde x}}_0(\cdot))y_0(\cdot),
    {{{N_0(\cdot, x_q(\cdot))}} \over {{d_q}}} \buildrel L^\infty \over \longrightarrow F_{0ux}({{\tilde x}}_0(\cdot))y_0(\cdot),

    both on T and, since v_q \buildrel L^1 \over \rightharpoonup v_0 on T ,

    \begin{align} {{ {{1} \over {2}}}} {J^{\prime\prime}}_0(z_0;(y_0, v_0)) = \lim\limits_{q\to\infty}{{{{{\cal K}}_0(z_q)}} \over {{d_q^2}}} + {{{1}} \over {{2}}}{\int_{t_0}^{t_1}} v_0^\ast(t)F_{0uu}({{\tilde x}}_0(t))v_0(t) dt. \end{align} (6)

    We have,

    \begin{align} \liminf\limits_{q\to\infty}{{{{{\cal E}}_0(z_q)}} \over {{d_q^2}}} \ge {{{1}} \over {{2}}}{\int_{t_0}^{t_1}} v_0^\ast(t)F_{0uu}({{\tilde x}}_0(t))v_0(t) dt. \end{align} (7)

    Indeed, by condition (a) of Lemma 3.1, we are able to choose \Upsilon\subset T measurable such that u_q \buildrel \hbox{u} \over \longrightarrow u_0 on \Upsilon . Since z_q is admissible, then recalling the definition of W_q given in the beginning of this section, as one readily verifies, W_q \buildrel \hbox{u} \over \longrightarrow 1 on \Upsilon . Moreover, for all t\in \Upsilon and q\in{{\bf N}} ,

    {{{1}} \over {{d_q^2}}}E_0(t, x_q(t), u_0(t), u_q(t)) = {{ {{1} \over {2}}}} v_q^\ast(t)R_q(t)v_q(t)

    where

    R_q(t): = 2\int_0^1 (1-\lambda)F_{0uu}(t, x_q(t), u_0(t)+\lambda[u_q(t)-u_0(t)])d\lambda.

    Clearly,

    R_q(\cdot) \buildrel \hbox{u} \over \longrightarrow R_0(\cdot): = F_{0uu}({{\tilde x}}_0(\cdot)) \hbox{ on $\Upsilon$}.

    By condition (ⅰ) of Theorem 2.1, R_0(t)\ge 0 (t\in \Upsilon) . Additionally, by condition (ⅲ)(a) of Theorem 2.1, for all q\in{{\bf N}} ,

    E_0(t, x_q(t), u_0(t), u_q(t))\ge0 \quad ({\hbox{a.e. in}\ T}),

    and so, by condition (c) of Lemma 3.1,

    \begin{eqnarray*} \liminf\limits_{q\to\infty}{{{{{\cal E}}_0(z_q)}} \over {{d_q^2}}} & = & \liminf\limits_{q\to\infty}{{{1}} \over {{d_q^2}}}{\int_{t_0}^{t_1}} E_0(t, x_q(t), u_0(t), u_q(t))dt \ge \liminf\limits_{q\to\infty}{{{1}} \over {{d_q^2}}} \int_\Upsilon E_0(t, x_q(t), u_0(t), u_q(t))dt \\ & = & {{{1}} \over {{2}}} \liminf\limits_{q\to\infty} \int_\Upsilon v_q^\ast(t)R_q(t)v_q(t) dt \ge {{{1}} \over {{2}}} \int_\Upsilon v_0^\ast(t)R_0(t)v_0(t)dt. \end{eqnarray*}

    As \Upsilon can be chosen to differ from T by a set of an arbitrarily small measure and the function

    t\mapsto v_0^\ast(t)R_0(t)v_0(t)

    belongs to L^1(T; {{\bf R}}) , this inequality holds when \Upsilon = T and this establishes (7). By (3) and (5)–(7),

    {{ {{1} \over {2}}}} {J^{\prime\prime}}_0(z_0;(y_0, v_0)) \le \lim\limits_{q\to\infty}{{{{{\cal K}}_0(z_q)}} \over {{d_q^2}}} + \liminf\limits_{q\to\infty}{{{{{\cal E}}_0(z_q)}} \over {{d_q^2}}} = \liminf\limits_{q\to\infty}{{{J_0(z_q)-J_0(z_0)}} \over {{d_q^2}}} \le 0.

    If (y_0, v_0) = (0, 0) , then

    \lim\limits_{q\to\infty}{{{{{\cal K}}_0(z_q)}} \over {{d_q^2}}} = 0

    and so, by condition (ⅲ)(b) of Theorem 2.1,

    {{ {{1} \over {2}}}}\delta\le\liminf\limits_{q\to\infty}{{{{{\cal E}}_0(z_q)}} \over {{d_q^2}}} \le 0,

    which contradicts the positivity of \delta .

    For all q\in{{\bf N}} , we have

    {{\dot y}}_q(t) = A_q(t)y_q(t) + B_q(t)v_q(t) \ ({\hbox{a.e. in}\ T}), \quad y_q(t_0) = 0,

    where

    A_q(t) = \int_0^1 f_x(t, x_0(t) + \lambda[x_q(t)-x_0(t)], u_0(t))d\lambda,
    B_q(t) = \int_0^1 f_u(t, x_q(t), u_0(t) + \lambda[u_q(t)-u_0(t)])d\lambda.

    Since

    A_q(\cdot) \buildrel \hbox{u} \over \longrightarrow A_0(\cdot): = f_x({{\tilde x}}_0(\cdot)), \quad B_q(\cdot) \buildrel \hbox{u} \over \longrightarrow B_0(\cdot): = f_u({{\tilde x}}_0(\cdot)),

    y_q \buildrel \hbox{u} \over \longrightarrow y_0 and v_q \buildrel L^1 \over \rightharpoonup v_0 all on \Upsilon , it follows that {{\dot y}}_q \buildrel L^1 \over \rightharpoonup A_0y_0+B_0y_0 on \Upsilon . By condition (b) of Lemma 3.1, {{\dot y}}_q \buildrel L^1 \over \rightharpoonup \zeta_0 = {{\dot y}}_0 on \Upsilon . Therefore,

    {{\dot y}}_0(t) = A_0(t)y_0(t) + B_0(t)v_0(t)\quad(t\in \Upsilon).

    As \Upsilon can be chosen to differ from T by a set of an arbitrarily small measure, then there cannot exist a subset of T of positive measure in which the functions y_0 and v_0 do not satisfy the differential equation {{\dot y}}_0(t) = A_0(t)y_0(t)+B_0(t)v_0(t) . Consequently,

    {{\dot y}}_0(t) = A_0(t)y_0(t) + B_0(t)v_0(t)\quad({\hbox{a.e. in}\ T})

    and (ⅰ) and (ⅱ) of our claim are proved.

    Finally, in order to obtain (ⅲ)–(ⅴ) of our claim it is enough to copy the proofs of [28] from Eqs (8)–(15).

    In this section we shall prove Theorem 2.2. We first state an auxiliary result which is an immediate consequence of Lemmas 3.1 and 3.2 of [30].

    In the following lemma we shall assume that we are given u_0\in L^1(T; {{\bf R}^m}) and a sequence \{u_q\} in L^1(T; {{\bf R}^m}) such that

    \lim\limits_{q\to\infty}D_2(u_q-u_0) = 0 \quad\hbox{and}\quad d_{2q}: = [2D_2(u_q-u_0)]^{1/2} > 0 \quad (q\in{{\bf N}}).

    For all q\in{{\bf N}} define

    v_{2q}: = {{{u_q-u_0}} \over {{d_{2q}}}}.

    Lemma 4.1

    {\rm\bf a.} For some v_{02}\in L^2(T; {{\bf R}^m}) and a subsequence of \{u_q\} , v_{2q} \buildrel L^1 \over \rightharpoonup v_{02} on T .

    {\rm\bf b.} Let A_q\in L^\infty(T; {{\bf R}}^{n\times n}) and B_q\in L^\infty(T; {{\bf R}}^{n\times m}) be matrix functions for which there exist constants m_0, m_1 > 0 such that \|A_q\|_\infty\le m_0 , \|B_q\|_\infty\le m_1 (q\in{{\bf N}}) , and for all q\in{{\bf N}} denote by Y_q the solution of the initial value problem

    {{\dot y}}(t) = A_q(t)y(t)+B_q(t)v_{2q}(t) \ ({{a.e.\; in}\ T}), \quad y(t_0) = 0.

    Then there exist \sigma_0\in L^2(T; {{\bf R}^n}) and a subsequence of \{z_q\} , such that {{\dot Y}}_q \buildrel L^1 \over \rightharpoonup \sigma_0 on T , and hence if Y_0(t): = \int_{t_0}^t\sigma_0(\tau)d\tau (t\in T) , then Y_q \buildrel \hbox{u} \over \longrightarrow Y_0 on T .

    Proof. As we made with the proof of Theorem 2.1, the proof of Theorem 2.2 will be made by contraposition, that is, we shall assume that for all \theta_1, \theta_2 > 0 , there exists an admissible process (x, u) such that

    \begin{align} \|(x, u)-(x_0, u_0)\| < \theta_1 \quad\hbox{and} \quad I(x, u) < I(x_0, u_0)+\theta_2 D_2(u-u_0). \end{align} (8)

    Once again, as we made with the proof of Theorem 2.1, (8) implies that for all \theta_1, \theta_2 > 0 , there exists (x, u) admissible with

    \begin{align} \|(x, u)-(x_0, u_0)\| < \theta_1 \quad\hbox{and}\quad J_0(x, u) < J_0(x_0, u_0)+\theta_2 D_2(u-u_0). \end{align} (9)

    Let z_0: = (x_0, u_0) . As in the proof of Theorem 2.1, for all admissible processes z = (x, u) ,

    J_0(z) = J_0(z_0)+{J^\prime}_0(z_0;z-z_0)+{{\cal K}}_0(z)+{{\cal E}}_0(z)

    where {{\cal E}}_0 and K_0 are given as in the proof of Theorem 2.1.

    Now, by (9), for all q\in{{\bf N}} there exists z_q: = (x_q, u_q) admissible such that

    \begin{align} \|z_q-z_0\| < {{{1}} \over {{q}}}, \quad J_0(z_q)-J_0(z_0) < {{{1}} \over {{q}}}D_2(u_q-u_0). \end{align} (10)

    Since z_q is admissible, the last inequality of (10) implies that u_q\not = u_0 and so

    d_{2q}: = [2D_2(u_q-u_0)]^{1/2} > 0 \quad (q\in{{\bf N}}).

    By the first relation of (10), we have

    \lim\limits_{q\to\infty}D_2(u_q-u_0) = 0.

    For all q\in{{\bf N}} , define v_{2q} as in Lemma 4.1 and

    Y_q: = {{{x_q-x_0}} \over {{d_{2q}}}} \quad\hbox{and}\quad W_{2q}: = [1+{{ {{1} \over {2}}}} V(u_q-u_0)]^{1/2}.

    By condition (a) of Lemma 4.1, there exist v_{02}\in L^2(T; {{\bf R}^m}) and a subsequence of \{z_q\} such that v_{2q} \buildrel L^1 \over \rightharpoonup v_{02} on T . As in the proof of Theorem 2.1, for all q\in{{\bf N}} ,

    \dot Y_q(t) = A_q(t)Y_q(t) + B_q(t)v_{2q}(t), \quad Y_q(t_0) = 0 \quad ({\hbox{a.e. in}\ T}).

    We have the existence of m_0, m_1 > 0 such that \|A_q\|_\infty\le m_0 and \|B_q\|_\infty\le m_1 (q\in{{\bf N}}) . By condition (b) of Lemma 4.1, there exist \sigma_0\in L^2(T; {{\bf R}^n}) and a subsequence of \{z_q\} such that, if Y_0(t): = \int_{t_0}^t \sigma_0(\tau)d\tau (t\in T) , then Y_q \buildrel \hbox{u} \over \longrightarrow Y_0 on T . We claim that

    ⅰ. {J^{\prime\prime}}_0(z_0;(Y_0, v_{02}))\le0 , (Y_0, v_{02})\not = (0, 0) .

    ⅱ. \dot Y_0(t) = f_x({{\tilde x}}_0(t))Y_0(t)+f_u({{\tilde x}}_0(t))v_{02}(t) ({\hbox{a.e. in}\ T}) , Y_0(t_i) = 0 (i = 0, 1) .

    ⅲ. {I^\prime}_i(z_0;(Y_0, v_{02}))\le0 (i\in i_a(z_0)) , {I^\prime}_j(z_0;(Y_0, v_{02})) = 0 (j = k+1, \ldots, K) .

    ⅳ. \varphi_{\alpha x}({{\tilde x}}_0(t))Y_0(t)+\varphi_{\alpha u}({{\tilde x}}_0(t))v_{02}(t)\le0 ({\hbox{a.e. in}\ T}, \, \alpha\in{{\cal I}}_a({{\tilde x}}_0(t))) .

    ⅴ. \varphi_{\beta x}({{\tilde x}}_0(t))Y_0(t)+\varphi_{\beta u}({{\tilde x}}_0(t))v_{02}(t) = 0 ({\hbox{a.e. in}\ T}, \, \beta\in S) .

    Indeed, for all q\in{{\bf N}} , we have

    {{{{{\cal K}}_0(z_q)}} \over {{d_{2q}^2}}} = {\int_{t_0}^{t_1}}\biggl \{ {{{M_0(t, x_q(t))}} \over {{d_{2q}^2}}} + v_{2q}^\ast(t) {{{N_0(t, x_q(t))}} \over {{d_{2q}}}} \biggr \}dt.

    Also, we have

    {{{M_0(\cdot, x_q(\cdot))}} \over {{d_{2q}^2}}} \buildrel L^\infty \over \longrightarrow {{ {{1} \over {2}}}} Y_0^\ast(\cdot) F_{0xx}({{\tilde x}}_0(\cdot))Y_0(\cdot),
    {{{N_0(\cdot, x_q(\cdot))}} \over {{d_{2q}}}} \buildrel L^\infty \over \longrightarrow F_{0ux}({{\tilde x}}_0(\cdot))Y_0(\cdot),

    both on T and, since v_{2q} \buildrel L^1 \over \rightharpoonup v_{02} on T ,

    \begin{align} {{ {{1} \over {2}}}} {J^{\prime\prime}}_0(z_0;(Y_0, v_{02})) = \lim\limits_{q\to\infty}{{{{{\cal K}}_0(z_q)}} \over {{d_{2q}^2}}} + {{{1}} \over {{2}}}{\int_{t_0}^{t_1}} v_{02}^\ast(t)F_{0uu}({{\tilde x}}_0(t))v_{02}(t) dt. \end{align} (11)

    Now, for all t\in T and q\in{{\bf N}} ,

    {{{1}} \over {{d_{2q}^2}}}E_0(t, x_q(t), u_0(t), u_q(t)) = {{ {{1} \over {2}}}} v_{2q}^\ast(t)R_q(t)v_{2q}(t)

    where

    R_q(t): = 2\int_0^1 (1-\lambda)F_{0uu}(t, x_q(t), u_0(t)+\lambda[u_q(t)-u_0(t)])d\lambda.

    Clearly,

    R_q(\cdot) \buildrel L^\infty \over \longrightarrow R_0(\cdot): = F_{0uu}({{\tilde x}}_0(\cdot)) \hbox{ on $T$}.

    Since \|z_q-z_0\|\to0 as q\to\infty , it follows that W_{2q} \buildrel L^\infty \over \longrightarrow 1 on T and, by condition (ⅰ) of Theorem 2.2, R_0(t)\ge 0 ({\hbox{a.e. in}\ T}) . Consequently,

    \begin{align} \liminf\limits_{q\to\infty}{{{{{\cal E}}_0(z_q)}} \over {{d_{2q}^2}}} \ge {{{1}} \over {{2}}} {\int_{t_0}^{t_1}} v_{02}^\ast(t)R_0(t)v_{02}(t)dt. \end{align} (12)

    On the other hand, since

    \dot\rho(t) = -H_x^\ast({{\tilde x}}_0(t), \rho(t), \mu(t)) \ ({\hbox{a.e. in}\ T}), \quad H_u^\ast({{\tilde x}}_0(t), \rho(t), \mu(t)) = 0 \ (t\in T),

    we have that {J^\prime}_0(z_0;(y, v)) = 0 for all (y, v)\in {{\cal X}}\times L^2(T; {{\bf R}^m}) . With this in mind, (10)–(12),

    {{ {{1} \over {2}}}} {J^{\prime\prime}}_0(z_0;(Y_0, v_{02})) \le \lim\limits_{q\to\infty}{{{{{\cal K}}_0(z_q)}} \over {{d_{2q}^2}}} + \liminf\limits_{q\to\infty}{{{{{\cal E}}_0(z_q)}} \over {{d_{2q}^2}}} = \liminf\limits_{q\to\infty}{{{J_0(z_q)-J_0(z_0)}} \over {{d_{2q}^2}}} \le 0.

    If (Y_0, v_{02}) = (0, 0) , then

    \lim\limits_{q\to\infty}{{{{{\cal K}}_0(z_q)}} \over {{d_{2q}^2}}} = 0

    and so, by condition (ⅲ) (\hbox{a}^\prime) of Theorem 2.2,

    {{ {{1} \over {2}}}}\delta\le\liminf\limits_{q\to\infty}{{{{{\cal E}}_0(z_q)}} \over {{d_{2q}^2}}} \le 0,

    which contradicts the positivity of \delta and this proves (ⅰ) of our claim.

    Now, we also claim that

    \dot Y_0(t) = f_x({{\tilde x}}_0(t))Y_0(t)+f_u({{\tilde x}}_0(t))v_{02}(t) \, \, ({\hbox{a.e. in}\ T}), \quad Y_0(t_i) = 0 \, \, (i = 0, 1).

    Indeed, the equalities Y_0(t_i) = 0 (i = 0, 1) follow from the definition of Y_q , the admissibility of z_q and the fact that Y_q \buildrel \hbox{u} \over \longrightarrow Y_0 on T . Also, observe that since Y_q \buildrel \hbox{u} \over \longrightarrow Y_0 ,

    A_q(\cdot) \buildrel L^\infty \over \longrightarrow A_0(\cdot): = f_x({{\tilde x}}_0(\cdot)),
    B_q(\cdot) \buildrel L^\infty \over \longrightarrow B_0(\cdot): = f_u({{\tilde x}}_0(\cdot)),

    and v_{2q} \buildrel L^1 \over \rightharpoonup v_{02} all on T , then \dot Y_q \buildrel L^1 \over \rightharpoonup A_0Y_0+B_0v_{02} on T . By condition (b) of Lemma 4.1, \dot Y_q \buildrel L^1 \over \rightharpoonup \sigma_0 = \dot Y_0 on T , which accordingly implies that

    \dot Y_0(t) = A_0(t)Y_0(t)+B_0(t)v_{02}(t) \quad ({\hbox{a.e. in}\ T})

    and our claim is proved.

    Finally, in order to prove (ⅲ)–(ⅴ) of our claim it is enough to copy the proofs given in [28] from Eqs (8)–(15) by replacing y_0 by Y_0 , v_0 by v_{02} and \Upsilon by T .

    In this article, we have provided sufficiency theorems for weak and strong minima in an optimal control problem of Lagrange with fixed end-points, nonlinear dynamics, inequality and equality isoperimetric restrictions and inequality and equality mixed time-state-control constraints. The sufficiency treatment studied in this paper does not need that the proposed optimal controls be continuous but only purely measurable. The sufficiency results not only provide local minima but they also measure the deviation between optimal and admissible costs by means of a functional playing a similar role of the square of the classical norm of the Banach space L^1 . Additionally, all the crucial sufficiency hypotheses are included in the theorems, in contrast, with other necessary and sufficiency theories which strongly depend upon some preliminary assumptions not embedded in the corresponding theorems of optimality. Finally, our sufficiency technique is self-contained because it is independent of some classical sufficient approaches involving Hamilton-Jacobi inequalities, matrix-valued Riccati equations, generalizations of Jacobi's theory appealing to extended notions of conjugate points or insertions of the original problem in some abstract Banach spaces.

    The author is thankful to Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México, for the support provided by the project PAPIIT-IN102220. Moreover, the author thanks the two anonymous referees for the encouraging suggestions made in their reviews.

    The author declares no conflict of interest.

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