This article implements an efficient analytical technique within three different operators to investigate the solutions of some fractional partial differential equations and their systems. The generalized schemes of the proposed method are derived for every targeted problem under the influence of each fractional derivative operator. The numerical examples of the non-homogeneous fractional Cauchy equation and three-dimensional Navier-Stokes equations are obtained using the new iterative transform method. The obtained results under different fractional derivative operators are found to be identical. The 2D and 3D plots have confirmed the close connection between the exact and obtained results. Moreover, the table shows the higher accuracy of the proposed method.
Citation: Qasim Khan, Anthony Suen, Hassan Khan, Poom Kumam. Comparative analysis of fractional dynamical systems with various operators[J]. AIMS Mathematics, 2023, 8(6): 13943-13983. doi: 10.3934/math.2023714
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This article implements an efficient analytical technique within three different operators to investigate the solutions of some fractional partial differential equations and their systems. The generalized schemes of the proposed method are derived for every targeted problem under the influence of each fractional derivative operator. The numerical examples of the non-homogeneous fractional Cauchy equation and three-dimensional Navier-Stokes equations are obtained using the new iterative transform method. The obtained results under different fractional derivative operators are found to be identical. The 2D and 3D plots have confirmed the close connection between the exact and obtained results. Moreover, the table shows the higher accuracy of the proposed method.
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 m≥3.
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×Rm∣φα(t,x,u)≤0(α∈R),φβ(t,x,u)=0(β∈S)} |
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 Υϵ⊂T with m(Υϵ)<ϵ such that {Λn} converges uniformly to Λ on T∖Υϵ. We will also denote uniform convergence by Λnu⟶Λ, almost uniform convergence by Λnau⟶Λ, strong convergence in Lp by ΛnLp⟶Λ and weak convergence in Lp by ΛnLp⇀Λ. 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 Υ⊂T measurable and some (Φ0,Ψ0)∈AC(T;Rn)×L∞(T;Rm), (Φq,Ψq)L∞⟶(Φ0,Ψ0) on Υ, then for all q∈N, b(⋅,Φq(⋅),Ψq(⋅)) is measurable on Υ and
b(⋅,Φq(⋅),Ψq(⋅))L∞⟶b(⋅,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:T→Rn absolutely continuous and u:T→Rm 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))dt≤0(i=1,…,k).Ij(x,u):=∫t1t0Lj(t,x(t),u(t))dt=0(j=k+1,…,K).(t,x(t),u(t))∈A(t∈T). |
Denote by X the space of all absolutely continuous functions mapping T to Rn and by Uc:=L∞(T;Rc) (c∈N). 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‖:=supt∈T|x(t)|, |
that is, if for any ϵ>0, I(x,u)≤I(y,v) for all admissible processes (y,v) satisfying ‖y−x‖<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 ‖x−x0‖∞ 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 ‖x−x0‖∞, ‖u−u0‖∞ 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×Um→R defined by
I0(x,u):=I(x,u)+K∑γ=1λγIγ(x,u)=∫t1t0L0(˜x(t))dt, |
where L0:T×Rn×Rm→R is given by
L0(t,x,u):=L(t,x,u)+K∑γ=1λγLγ(t,x,u). |
∙ For all (t,x,u,ρ,μ)∈T×Rn×Rm×Rn×Rs, set
H(t,x,u,ρ,μ):=ρ∗f(t,x,u)−L0(t,x,u)−μ∗φ(t,x,u). |
Given ρ∈X and μ∈Us 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)ξ1−ρ∗(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
J′0((x,u);(y,v)):=∫t1t0{F0x(˜x(t))y(t)+F0u(˜x(t))v(t)}dt, |
I′γ((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
J′′0((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):=y∗F0xx(˜x(t))y+2y∗F0xu(˜x(t))v+v∗F0uu(˜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)(v−u). |
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)(v−u). |
∙ 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)1/2−1 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):={α∈R∣φα(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,…,k∣Ii(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).I′i((x,u);(y,v))≤0(i∈ia(x,u)),I′j((x,u);(y,v))=0(j=k+1,…,K).φαx(˜x(t))y(t)+φαu(˜x(t))v(t)≤0(a.e. in T,α∈Ia(˜x(t))).φβx(˜x(t))y(t)+φβu(˜x(t))v(t)=0(a.e. in T,β∈S). |
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 ρ∈X, μ∈Us with μα(t)≥0 and μα(t)φα(˜x0(t))=0 (α∈R,t∈T), two positive numbers δ,ϵ, and multipliers λ1,…,λK with λi≥0 and λiIi(x0,u0)=0 (i=1,…,k) such that
˙ρ(t)=−H∗x(˜x0(t),ρ(t),μ(t)) (a.e.in T), |
H∗u(˜x0(t),ρ(t),μ(t))=0 (t∈T), |
and the following holds:
(ⅰ) Huu(˜x0(t),ρ(t),μ(t))≤0 (a.e.in T).
(ⅱ) J′′0((x0,u0);(y,v))>0 for all (y,v)≠(0,0), (y,v)∈Y(x0,u0).
(ⅲ) If (x,u) is admissible with ‖x−x0‖<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))dt≥δmax{∫t1t0V(˙x(t)−˙x0(t))dt,∫t1t0V(u(t)−u0(t))dt}.
c. ∫t1t0E0(t,x(t),u0(t),u(t))dt≥δ|∫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−x0‖<θ1,
I(x,u)≥I(x0,u0)+θ2D(x−x0,u−u0). |
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 ρ∈X, μ∈Us with μα(t)≥0 and μα(t)φα(˜x0(t))=0 (α∈R,t∈T), two positive numbers δ,ϵ, and multipliers λ1,…,λK with λi≥0 and λiIi(x0,u0)=0 (i=1,…,k) such that
˙ρ(t)=−H∗x(˜x0(t),ρ(t),μ(t)) (a.e.in T), |
H∗u(˜x0(t),ρ(t),μ(t))=0 (t∈T), |
and the following holds:
(ⅰ) Huu(˜x0(t),ρ(t),μ(t))≤0 (a.e.in T).
(ⅱ) J′′0((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))dt≥δ∫t1t0V(u(t)−u0(t))dt.
b′. ∫t1t0E0(t,x(t),u0(t),u(t))dt≥δ|∫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(u−u0). |
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]×R→R 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)}dt≤0.(t,x(t),u(t))∈A (t∈[0,1]) |
where
A:={(t,x,u)∈[0,1]×R×R2∣u1≥0, (u2−u02(t))2≤1, 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)=(u2−u02(t))2−1,φ3(t,x,u)=u22−1. |
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+12ρx−sinh(u1)−u21cos(2πu2)+x2−λ1[14x2+xu1u2]+μ1u1−μ2[(u2−u02(t))2−1]−μ3[u22−1], |
Hx(t,x,u,ρ,μ)=12ρ+2x−λ1[12x+u1u2], |
Hu(t,x,u,ρ,μ)=(ρu2−cosh(u1)−2u1cos(2πu2)−λ1xu2+μ1ρu1+2πu21sin(2πu2)−λ1xu1−2μ2(u2−u02(t))−2μ3u2)∗. |
Therefore, if we set ρ≡0, μ1≡1, μ2=μ3≡0 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) (t∈T), |
and hence the first order sufficient conditions involving the Hamiltonian of problem (P) are verified. Moreover, if we set R:={1,2}, observe that
λ1≥0,λ1I1(x0,u0)=0, |
μα(t)≥0,μα(t)φα(˜x0(t))=0(α∈R,t∈T). |
Additionally, Ia(˜x0(⋅))≡{1} is constant on T. Also, it is readily seen that for all t∈T,
Huu(˜x0(t),ρ(t),μ(t))=(−2000), |
and so condition (ⅰ) of Theorem 2.1 is satisfied. Observe that, for all t∈T,
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)−x2−u1, |
and so, for all t∈T,
F0xx(˜x0(t))=−2,F0xu(˜x0(t))=(0,0),F0uu(˜x0(t))=(2000). |
Consequently, we have
12J′′0((x0,u0);(y,v))=∫10{v21(t)−y2(t)}dt=∫10{(˙y(t)−12y(t))2−y2(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 t∈T,
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
u−u0=(u1−u01,u2−u02)=(u1,u02−u02)=(u1,0) |
and so, if (x,u) is admissible,
∫10E0(t,x(t),u0(t),u(t))dt≥∫10u21(t)dt≥∫10V(u1(t))dt=∫10V(u(t)−u0(t))dt. |
Also, if (x,u) is admissible, then
∫10E0(t,x(t),u0(t),u(t))dt≥∫10u21(t)dt=∫10{(u1(t)u2(t)+12x(t))2−x(t)u1(t)u2(t)−14x2(t)}dt=∫10{˙x2(t)−x(t)u1(t)u2(t)−14x2(t)}dt≥∫10˙x2(t)dt≥∫10V(˙x(t)−˙x0(t))dt. |
Therefore, if (x,u) is admissible, then
∫10E0(t,x(t),u0(t),u(t))dt≥max{∫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|=|∫10−12x2(t)dt|=12∫10x2(t)dt≤∫10˙x2(t)dt=∫10(u1(t)u2(t)+12x(t))2dt=∫10u21(t)dt+∫10{x(t)u1(t)u2(t)+14x2(t)}dt≤∫10u21(t)dt+∫10x(t)˙x(t)dt=∫10u21(t)dt≤∫10E0(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))}dt≤0.(t,x(t),u(t))∈A (t∈[0,1]) |
where
A:={(t,x,u)∈[0,1]×R×R2∣sin(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, |
L1(t,x,u)=sin(u1)−sinh(u1+u1x3), |
L0(t,x,u)=sinh(u1+u1x3)+12u21cos(2πu2)−cosh(x)+1+λ1[sin(u1)−sinh(u1+u1x3)], |
φ1(t,x,u)=−sin(u1),φ2(t,x,u)=u22−1. |
Clearly, L, L1, f and φ=(φ1,φ2) satisfy the hypotheses imposed in the statement of the problem.
Let u02:T→R be any measurable function whose codomain belongs to the set {−1,1}.
Clearly, the process (x0,u0)=(x0,u01,u02)≡(0,0,u02) is admissible.
Moreover,
H(t,x,u,ρ,μ)=ρu1u2+ρx−sinh(u1+u1x3)−12u21cos(2πu2)+cosh(x)−1−λ1[sin(u1)−sinh(u1+u1x3)]+μ1sin(u1)−μ2[u22−1], |
Hx(t,x,u,ρ,μ)=ρ−3x2u1cosh(u1+u1x3)+sinh(x)+3λ1x2u1cosh(u1+u1x3), |
Hu1(t,x,u,ρ,μ)=ρu2−[1+x3]cosh(u1+u1x3)−u1cos(2πu2)−λ1[cos(u1)−{1+x3}cosh(u1+u1x3)]+μ1cos(u1), |
Hu2(t,x,u,ρ,μ)=ρu1+πu21sin(2πu2)−2μ2u2. |
Therefore, if we set ρ≡0, μ1≡1, μ2≡0 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) (t∈T), |
and hence the first order sufficient conditions involving the Hamiltonian of problem (P) are verified. Additionally, observe that
λ1≥0,λ1I1(x0,u0)=0, |
μ1(t)≥0,μ1(t)φ1(˜x0(t))=0(t∈T). |
Also, Ia(˜x0(⋅))≡{1} is constant on T. Moreover, it is readily seen that for all t∈T,
Huu(˜x0(t),ρ(t),μ(t))=(−1000), |
and so condition (ⅰ) of Theorem 2.2 is satisfied. Observe that, for all t∈T,
fx(˜x0(t))=1,fu(˜x0(t))=(u02(t),0),L1x(˜x0(t))=0,L1u(˜x0(t))=(0,0), |
φ1x(˜x0(t))=0,φ1u(˜x0(t))=(−1,0),φ2x(˜x0(t))=0,φ2u(˜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)=y(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). |
Also, 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+u1x3)+12u21cos(2πu2)−cosh(x)+1−sin(u1), |
and so, for all t∈T,
F0xx(˜x0(t))=−1,F0xu(˜x0(t))=(0,0),F0uu(˜x0(t))=(1000). |
Consequently, we have
J′′0((x0,u0);(y,v))=∫10{v21(t)−y2(t)}dt=∫10{(˙y(t)−y(t))2−y2(t)}dt=∫10{˙y2(t)−2y(t)˙y(t)}dt=∫10˙y2(t)dt>0 |
for all (y,v)≠(0,0), (y,v)∈Y(x0,u0). Hence, condition (ⅱ) of Theorem 2.2 is verified.
Additionally, observe that for any ϵ∈(0,1), all (x,u) admissible satisfying ‖(x,u)−(x0,u0)‖<ϵ and all t∈T,
E0(t,x(t),u0(t),u(t))=sinh(u1(t)+u1(t)x3(t))+12u21(t)cos(2πu2(t))−sin(u1(t))=sinh(u1(t)+u1(t)x3(t))+12u21(t)cos(2πu02(t))−sin(u1(t))=sinh(u1(t)+u1(t)x3(t))+12u21(t)−sin(u1(t)). |
Therefore, for any ϵ∈(0,1) and all (x,u) admissible satisfying ‖(x,u)−(x0,u0)‖<ϵ,
∫10E0(t,x(t),u0(t),u(t))dt=∫10{sinh(u1(t)+u1(t)x3(t))−sin(u1(t))+12u21(t)}dt≥∫1012u21(t)dt≥∫10V(u1(t))dt=∫10V(u(t)−u0(t))dt, |
and hence condition (ⅲ)(a′) of Theorem 2.2 is satisfied for any ϵ∈(0,1) and δ=1.
Finally, if (x,u) is admissible, note that
∫10E0(t,x(t),u0(t),u(t))dt≥|∫10{sinh(u1(t)+u1(t)x3(t))−sin(u1(t))}dt|=|∫10E1(t,x(t),u0(t),u(t))dt|, |
implying that condition (ⅲ)(b′) of Theorem 2.2 holds for any ϵ>0 and δ=1. By Theorem 2.2, (x0,u0) 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 z0:=(x0,u0)∈X×L1(T;Rm) and a subsequence {zq:=(xq,uq)} in X×L1(T;Rm) such that
limq→∞D(zq−z0)=0anddq:=[2D(zq−z0)]1/2>0(q∈N). |
For all q∈N, set
yq:=xq−x0dqandvq:=uq−u0dq. |
For all q∈N, define
Wq:=max{W1q,W2q} |
where
W1q:=[1+12V(˙xq−˙x0)]1/2andW2q:=[1+12V(uq−u0)]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
a. For some v0∈L2(T;Rm) and some subsequence of {zq}, vqL1⇀v0 on T. Even more, uqau⟶u0 on T.
b. There exist ζ0∈L2(T;Rn), ˉy0∈Rn, and some subsequence of {zq}, such that ˙yqL1⇀ζ0 on T. Moreover, if y0(t):=ˉy0+∫tt0ζ0(τ)dτ (t∈T), then yqu⟶y0 on T.
c. Let Υ⊂T be measurable and suppose that Wqu⟶1 on Υ. Let Rq,R0∈L∞(Υ;Rm×m), assume that Rqu⟶R0 on Υ, R0(t)≥0 (t∈Υ), and let v0 be the function considered in condition (a) of Lemma 3.1. Then,
lim infq→∞∫Υv∗q(t)Rq(t)vq(t)dt≥∫Υv∗0(t)R0(t)v0(t)dt. |
Proof. The proof of Theorem 2.1 will be made by contraposition, that is, we shall assume that for all θ1,θ2>0, there exists an admissible process (x,u) such that
‖x−x0‖<θ1andI(x,u)<I(x0,u0)+θ2D(x−x0,u−u0). | (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
μα(t)≥0 (α∈R,t∈T)andλi≥0 (i=1,…,k), |
if (x,u) is admissible, then I(x,u)≥J0(x,u). Also, since
μα(t)φα(˜x0(t))=0 (α∈R,t∈T)andλiIi(x0,u0)=0 (i=1,…,k), |
then I(x0,u0)=J0(x0,u0). Thus, (1) implies that for all θ1,θ2>0, there exists (x,u) admissible with ‖x−x0‖<θ1 and
J0(x,u)<J0(x0,u0)+θ2D(x−x0,u−u0). | (2) |
Let z0:=(x0,u0). Note that, for all admissible processes z=(x,u),
J0(z)=J0(z0)+J′0(z0;z−z0)+K0(z)+E0(z) | (3) |
where
E0(x,u):=∫t1t0E0(t,x(t),u0(t),u(t))dt, |
K0(x,u):=∫t1t0{M0(t,x(t))+[u∗(t)−u∗0(t)]N0(t,x(t))}dt, |
and the functions M0 and N0 are given by
M0(t,y):=F0(t,y,u0(t))−F0(˜x0(t))−F0x(˜x0(t))(y−x0(t)), |
N0(t,y):=F∗0u(t,y,u0(t))−F∗0u(˜x0(t)). |
We have,
M0(t,y)=12[y∗−x∗0(t)]P0(t,y)(y−x0(t)),N0(t,y)=Q0(t,y)(y−x0(t)), |
where
P0(t,y):=2∫10(1−λ)F0xx(t,x0(t)+λ[y−x0(t)],u0(t))dλ, |
Q0(t,y):=∫10F0ux(t,x0(t)+λ[y−x0(t)],u0(t))dλ. |
Now, as in [28], choose ν>0 such that for all z=(x,u) admissible with ‖x−x0‖<1,
|K0(x,u)|≤ν‖x−x0‖[1+D(z−z0)]. | (4) |
Now, by (2), for all q∈N there exists zq:=(xq,uq) admissible such that
‖xq−x0‖<ϵ,‖xq−x0‖<1q,J0(zq)−J0(z0)<1qD(zq−z0). | (5) |
The last inequality of (5) implies that zq≠z0 and so for all q∈N,
dq:=[2D(zq−z0)]1/2>0. |
Since
˙ρ(t)=−H∗x(˜x0(t),ρ(t),μ(t)) (a.e. in T),H∗u(˜x0(t),ρ(t),μ(t))=0 (t∈T), |
it follows that J′0(z0;(y,v))=0 for all (y,v)∈X×L2(T;Rm). With this in mind, by (3), condition (ⅲ)(b) of Theorem 2.1, (4) and (5),
J0(zq)−J0(z0)=K0(zq)+E0(zq)≥−ν‖xq−x0‖+D(zq−z0)(δ−ν‖xq−x0‖). |
By (5), for all q∈N,
D(zq−z0)(δ−1q−νq)<νq |
and hence
limq→∞D(zq−z0)=0. |
For all q∈N, define
yq:=xq−x0dqandvq:=uq−u0dq. |
By condition (a) of Lemma 3.1, there exist v0∈L2(T;Rm) and a subsequence of {zq} such that vqL1⇀v0 on T. By condition (b) of Lemma 3.1, there exist ζ0∈L2(T;Rn), ˉy0∈Rn and a subsequence of {zq} such that, if for all t∈T, y0(t):=ˉy0+∫tt0ζ0(τ)dτ, then yqu⟶y0 on T.
We claim that
ⅰ. J′′0(z0;(y0,v0))≤0, (y0,v0)≠(0,0).
ⅱ. ˙y0(t)=fx(˜x0(t))y0(t)+fu(˜x0(t))v0(t) (a.e. in T), y0(ti)=0 (i=0,1).
ⅲ. I′i(z0;(y0,v0))≤0 (i∈ia(z0)), I′j(z0;(y0,v0))=0 (j=k+1,…,K).
ⅳ. φαx(˜x0(t))y0(t)+φαu(˜x0(t))v0(t)≤0 (a.e. in T,α∈Ia(˜x0(t))).
ⅴ. φβx(˜x0(t))y0(t)+φβu(˜x0(t))v0(t)=0 (a.e. in T,β∈S).
Indeed, the equalities y0(ti)=0 (i=0,1) follow from the definition of yq, the admissibility of zq and the fact that yqu⟶y0 on T.
For all q∈N, we have
K0(zq)d2q=∫t1t0{M0(t,xq(t))d2q+v∗q(t)N0(t,xq(t))dq}dt. |
By condition (b) of Lemma 3.1,
M0(⋅,xq(⋅))d2qL∞⟶12y∗0(⋅)F0xx(˜x0(⋅))y0(⋅), |
N0(⋅,xq(⋅))dqL∞⟶F0ux(˜x0(⋅))y0(⋅), |
both on T and, since vqL1⇀v0 on T,
12J′′0(z0;(y0,v0))=limq→∞K0(zq)d2q+12∫t1t0v∗0(t)F0uu(˜x0(t))v0(t)dt. | (6) |
We have,
lim infq→∞E0(zq)d2q≥12∫t1t0v∗0(t)F0uu(˜x0(t))v0(t)dt. | (7) |
Indeed, by condition (a) of Lemma 3.1, we are able to choose Υ⊂T measurable such that uqu⟶u0 on Υ. Since zq is admissible, then recalling the definition of Wq given in the beginning of this section, as one readily verifies, Wqu⟶1 on Υ. Moreover, for all t∈Υ and q∈N,
1d2qE0(t,xq(t),u0(t),uq(t))=12v∗q(t)Rq(t)vq(t) |
where
Rq(t):=2∫10(1−λ)F0uu(t,xq(t),u0(t)+λ[uq(t)−u0(t)])dλ. |
Clearly,
Rq(⋅)u⟶R0(⋅):=F0uu(˜x0(⋅)) on Υ. |
By condition (ⅰ) of Theorem 2.1, R0(t)≥0 (t∈Υ). Additionally, by condition (ⅲ)(a) of Theorem 2.1, for all q∈N,
E0(t,xq(t),u0(t),uq(t))≥0(a.e. in T), |
and so, by condition (c) of Lemma 3.1,
lim infq→∞E0(zq)d2q=lim infq→∞1d2q∫t1t0E0(t,xq(t),u0(t),uq(t))dt≥lim infq→∞1d2q∫ΥE0(t,xq(t),u0(t),uq(t))dt=12lim infq→∞∫Υv∗q(t)Rq(t)vq(t)dt≥12∫Υv∗0(t)R0(t)v0(t)dt. |
As Υ can be chosen to differ from T by a set of an arbitrarily small measure and the function
t↦v∗0(t)R0(t)v0(t) |
belongs to L1(T;R), this inequality holds when Υ=T and this establishes (7). By (3) and (5)–(7),
12J′′0(z0;(y0,v0))≤limq→∞K0(zq)d2q+lim infq→∞E0(zq)d2q=lim infq→∞J0(zq)−J0(z0)d2q≤0. |
If (y0,v0)=(0,0), then
limq→∞K0(zq)d2q=0 |
and so, by condition (ⅲ)(b) of Theorem 2.1,
12δ≤lim infq→∞E0(zq)d2q≤0, |
which contradicts the positivity of δ.
For all q∈N, we have
˙yq(t)=Aq(t)yq(t)+Bq(t)vq(t) (a.e. in T),yq(t0)=0, |
where
Aq(t)=∫10fx(t,x0(t)+λ[xq(t)−x0(t)],u0(t))dλ, |
Bq(t)=∫10fu(t,xq(t),u0(t)+λ[uq(t)−u0(t)])dλ. |
Since
Aq(⋅)u⟶A0(⋅):=fx(˜x0(⋅)),Bq(⋅)u⟶B0(⋅):=fu(˜x0(⋅)), |
yqu⟶y0 and vqL1⇀v0 all on Υ, it follows that ˙yqL1⇀A0y0+B0y0 on Υ. By condition (b) of Lemma 3.1, ˙yqL1⇀ζ0=˙y0 on Υ. Therefore,
˙y0(t)=A0(t)y0(t)+B0(t)v0(t)(t∈Υ). |
As Υ 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 y0 and v0 do not satisfy the differential equation ˙y0(t)=A0(t)y0(t)+B0(t)v0(t). Consequently,
˙y0(t)=A0(t)y0(t)+B0(t)v0(t)(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 u0∈L1(T;Rm) and a sequence {uq} in L1(T;Rm) such that
limq→∞D2(uq−u0)=0andd2q:=[2D2(uq−u0)]1/2>0(q∈N). |
For all q∈N define
v2q:=uq−u0d2q. |
Lemma 4.1
a. For some v02∈L2(T;Rm) and a subsequence of {uq}, v2qL1⇀v02 on T.
b. Let Aq∈L∞(T;Rn×n) and Bq∈L∞(T;Rn×m) be matrix functions for which there exist constants m0,m1>0 such that ‖Aq‖∞≤m0, ‖Bq‖∞≤m1 (q∈N), and for all q∈N denote by Yq the solution of the initial value problem
˙y(t)=Aq(t)y(t)+Bq(t)v2q(t) (a.e.in T),y(t0)=0. |
Then there exist σ0∈L2(T;Rn) and a subsequence of {zq}, such that ˙YqL1⇀σ0 on T, and hence if Y0(t):=∫tt0σ0(τ)dτ (t∈T), then Yqu⟶Y0 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 θ1,θ2>0, there exists an admissible process (x,u) such that
‖(x,u)−(x0,u0)‖<θ1andI(x,u)<I(x0,u0)+θ2D2(u−u0). | (8) |
Once again, as we made with the proof of Theorem 2.1, (8) implies that for all θ1,θ2>0, there exists (x,u) admissible with
‖(x,u)−(x0,u0)‖<θ1andJ0(x,u)<J0(x0,u0)+θ2D2(u−u0). | (9) |
Let z0:=(x0,u0). As in the proof of Theorem 2.1, for all admissible processes z=(x,u),
J0(z)=J0(z0)+J′0(z0;z−z0)+K0(z)+E0(z) |
where E0 and K0 are given as in the proof of Theorem 2.1.
Now, by (9), for all q∈N there exists zq:=(xq,uq) admissible such that
‖zq−z0‖<1q,J0(zq)−J0(z0)<1qD2(uq−u0). | (10) |
Since zq is admissible, the last inequality of (10) implies that uq≠u0 and so
d2q:=[2D2(uq−u0)]1/2>0(q∈N). |
By the first relation of (10), we have
limq→∞D2(uq−u0)=0. |
For all q∈N, define v2q as in Lemma 4.1 and
Yq:=xq−x0d2qandW2q:=[1+12V(uq−u0)]1/2. |
By condition (a) of Lemma 4.1, there exist v02∈L2(T;Rm) and a subsequence of {zq} such that v2qL1⇀v02 on T. As in the proof of Theorem 2.1, for all q∈N,
˙Yq(t)=Aq(t)Yq(t)+Bq(t)v2q(t),Yq(t0)=0(a.e. in T). |
We have the existence of m0,m1>0 such that ‖Aq‖∞≤m0 and ‖Bq‖∞≤m1 (q∈N). By condition (b) of Lemma 4.1, there exist σ0∈L2(T;Rn) and a subsequence of {zq} such that, if Y0(t):=∫tt0σ0(τ)dτ (t∈T), then Yqu⟶Y0 on T. We claim that
ⅰ. J′′0(z0;(Y0,v02))≤0, (Y0,v02)≠(0,0).
ⅱ. ˙Y0(t)=fx(˜x0(t))Y0(t)+fu(˜x0(t))v02(t) (a.e. in T), Y0(ti)=0 (i=0,1).
ⅲ. I′i(z0;(Y0,v02))≤0 (i∈ia(z0)), I′j(z0;(Y0,v02))=0 (j=k+1,…,K).
ⅳ. φαx(˜x0(t))Y0(t)+φαu(˜x0(t))v02(t)≤0 (a.e. in T,α∈Ia(˜x0(t))).
ⅴ. φβx(˜x0(t))Y0(t)+φβu(˜x0(t))v02(t)=0 (a.e. in T,β∈S).
Indeed, for all q∈N, we have
K0(zq)d22q=∫t1t0{M0(t,xq(t))d22q+v∗2q(t)N0(t,xq(t))d2q}dt. |
Also, we have
M0(⋅,xq(⋅))d22qL∞⟶12Y∗0(⋅)F0xx(˜x0(⋅))Y0(⋅), |
N0(⋅,xq(⋅))d2qL∞⟶F0ux(˜x0(⋅))Y0(⋅), |
both on T and, since v2qL1⇀v02 on T,
12J′′0(z0;(Y0,v02))=limq→∞K0(zq)d22q+12∫t1t0v∗02(t)F0uu(˜x0(t))v02(t)dt. | (11) |
Now, for all t∈T and q∈N,
1d22qE0(t,xq(t),u0(t),uq(t))=12v∗2q(t)Rq(t)v2q(t) |
where
Rq(t):=2∫10(1−λ)F0uu(t,xq(t),u0(t)+λ[uq(t)−u0(t)])dλ. |
Clearly,
Rq(⋅)L∞⟶R0(⋅):=F0uu(˜x0(⋅)) on T. |
Since ‖zq−z0‖→0 as q→∞, it follows that W2qL∞⟶1 on T and, by condition (ⅰ) of Theorem 2.2, R0(t)≥0 (a.e. in T). Consequently,
lim infq→∞E0(zq)d22q≥12∫t1t0v∗02(t)R0(t)v02(t)dt. | (12) |
On the other hand, since
˙ρ(t)=−H∗x(˜x0(t),ρ(t),μ(t)) (a.e. in T),H∗u(˜x0(t),ρ(t),μ(t))=0 (t∈T), |
we have that J′0(z0;(y,v))=0 for all (y,v)∈X×L2(T;Rm). With this in mind, (10)–(12),
12J′′0(z0;(Y0,v02))≤limq→∞K0(zq)d22q+lim infq→∞E0(zq)d22q=lim infq→∞J0(zq)−J0(z0)d22q≤0. |
If (Y0,v02)=(0,0), then
limq→∞K0(zq)d22q=0 |
and so, by condition (ⅲ)(a′) of Theorem 2.2,
12δ≤lim infq→∞E0(zq)d22q≤0, |
which contradicts the positivity of δ and this proves (ⅰ) of our claim.
Now, we also claim that
˙Y0(t)=fx(˜x0(t))Y0(t)+fu(˜x0(t))v02(t)(a.e. in T),Y0(ti)=0(i=0,1). |
Indeed, the equalities Y0(ti)=0 (i=0,1) follow from the definition of Yq, the admissibility of zq and the fact that Yqu⟶Y0 on T. Also, observe that since Yqu⟶Y0,
Aq(⋅)L∞⟶A0(⋅):=fx(˜x0(⋅)), |
Bq(⋅)L∞⟶B0(⋅):=fu(˜x0(⋅)), |
and v2qL1⇀v02 all on T, then ˙YqL1⇀A0Y0+B0v02 on T. By condition (b) of Lemma 4.1, ˙YqL1⇀σ0=˙Y0 on T, which accordingly implies that
˙Y0(t)=A0(t)Y0(t)+B0(t)v02(t)(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 y0 by Y0, v0 by v02 and Υ 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 L1. 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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