Covering cross-polytopes with smaller homothetic copies

1.   Introduction

Necessary and sufficient conditions for optimality play a crucial role in solving problems in the calculus of variations. The main necessary conditions for such problems are the well-known conditions of Euler, Weierstrass, Legendre and Jacobi. In many cases, depending on the smoothness of the assumptions, Euler's necessary condition corresponds to a second order differential equation which restricts solutions to lie in a family of trajectories with certain uniformity properties. Also, the necessary condition of Jacobi cannot be applied when the extremal has corners and is not nonsingular. This is an unfortunate feature since, in general, the admissible arcs or trajectories which are candidates for solving the problem are neither nonsingular nor smooth.

On the other hand, one fundamental aspect of the theory of sufficient conditions for optimality consists in slightly strengthening the necessary conditions. Concretely, if an admissible arc satisfies the strengthened conditions of Euler, Legendre and Jacobi, then it is a strict weak minimum. Additionally, if this admissible arc also satisfies the strengthened condition of Weierstrass, then it is a strict strong minimum. Some of the techniques used to obtain sufficiency include the construction of a Mayer field on which the extremals are independent of the path with respect to an invariant integral commonly called the Hilbert integral, the existence of a symmetric solution of the matrix Riccati inequality associated with the problem, a verification function satisfying the Hamilton-Jacobi equation, a quadratic function that satisfies a Hamilton-Jacobi inequality, the nonexistence of conjugate points on the underlying half-closed time interval, or the incorporation of some convexity arguments on the functions delimiting the calculus of variations problem (see for example ^{[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]} and references therein).

It is important to mention that the smoothness and the nonsingularity assumptions are crucial in the sufficiency theories mentioned above. In other words, the classical sufficiency theory in the calculus of variations, in general, may not give a response when the extremal under consideration is singular or has corners. In fact, as mentioned in ^[6], there is a gap between the set of necessary and sufficient conditions and [6,Section 3.7] is entirely devoted to the study of some problems for which the nonsingularity assumption fails. There, one finds a method which is only applicable to particular examples and may not hold in general, since "although this algorithm sheds light on the theory, it provides no panacea. Indeed there are no panaceas." Additionally, we refer the reader to ^[13], where the importance of the nonsingularity assumption in the classical calculus of variations sufficiency theory is fully explained.

In this paper we derive two new sufficiency results which provide sufficient conditions for strong local minima in certain classes of parametric and nonparametric calculus of variations problems of Bolza with variable or free end-points, inequality and equality nonlinear isoperimetric constrains, and nonlinear mixed pointwise inequality and equality constraints. The main novelty of our new sufficient theorems concerns their applicability to cases in which the extremals under consideration may be singular and nonsmooth, that is, the strengthened condition of Legendre and the continuity of the derivative of the proposed extremal are no longer required. More precisely, given an admissible extremal whose derivative is not continuous nor piecewise continuous but only essentially bounded, the elements comprising the new sufficiency theorems are the classical transversality condition, a crucial inequality which arises from the original algorithm used to prove the main result of the article, the necessary condition of Legendre, but not its strengthened version, the positivity of the second variation over the set of all nonnul admissible variations, and three refined Weierstrass conditions which are related to the functions delimiting the problems.

Another distinguishing characteristic of the main sufficiency result of the nonparametric calculus of variations problem presented in this paper is the fact that the initial and final end-points of the states are completely free, that is, they are not only variable end-points but they may belong to any set which is not necessarily a smooth manifold described by some functions which usually involve some type of equality or inequality conditions.

The paper is organized as follows. In Section 2 we pose the parametric calculus of variations problem we shall deal with together with some basic definitions and the statement of the main result of the article. In Section 3 we enunciate the nonparametric calculus of variations problem we shall study together with some basic definitions and a corollary which is also one of the main results of the paper. Section 4 is devoted to state two auxiliary lemmas in which the proof of the theorem is strongly based. Section 5 is dedicated to the proof of the main theorem of the article. In Section 6 we prove the lemmas given in Section 4 and, in the final section, we provide an example which shows how the sufficient theory developed in this paper widens the range of applicability of the classical calculus of variations theory.

2.   Statement of a parametric problem of Bolza and the main result

Suppose we are given an interval $T: = [t_0, t_1]$ in ${\bf R}$, and functions $l(b)\colon{\bf R}^p\to{\bf R}$, $l_\gamma(b)\colon{\bf R}^p\to{\bf R}$ $(\gamma = 1, \ldots, K)$, $\Psi_i(b)\colon{\bf R}^p\to{\bf R}^n$ $(i = 0, 1)$, $L(t, x, {\dot x}):\; T \times {\bf R}^n \times {\bf R}^n\to{\bf R}$, $L_\gamma(t, x, {\dot x}):\; T \times {\bf R}^n \times {\bf R}^n\to{\bf R}$ $(\gamma = 1, \ldots, K)$ and $\varphi(t, x, {\dot x}):\; T \times {\bf R}^n \times {\bf R}^n\to{\bf R}^s$. Set

where $R: = \{1, \ldots, r\}$ and $S: = \{r+1, \ldots, s\}$ $(r = 0, 1, \ldots, s)$. If $r = 0$ then $R = \emptyset$ and we disregard statements involving $\varphi_\alpha$. Similarly, if $r = s$ then $S = \emptyset$ and we disregard statements involving $\varphi_\beta$.

It will be assumed throughout the paper that $L$, $L_\gamma$ $(\gamma = 1, \ldots, K)$ and $\varphi = (\varphi_1, \ldots, \varphi_s)$ have first and second derivatives with respect to $x$ and ${\dot x}$. Moreover, we shall assume that the functions $l$, $l_\gamma$ $(\gamma = 1, \ldots, K)$ and $\Psi_i$ $(i = 0, 1)$ are of class $C^2$ on ${\bf R}^p$. Also, if we denote by $c(t, x, {\dot x})$ either $L(t, x, {\dot x})$, $L_\gamma(t, x, {\dot x})$ $(\gamma = 1, \ldots, K)$, $\varphi(t, x, {\dot x})$ or any of its partial derivatives of order less than or equal to two with respect to $x$ and ${\dot x}$, we shall assume that if ${\cal C}$ is any bounded subset of $T \times {\bf R}^n \times {\bf R}^n$, then $|c({\cal C})|$ is a bounded subset of ${\bf R}$. Additionally, we shall assume that if $\{(\Gamma_q, \Lambda_q)\}$ is any sequence in ${AC}(T; {\bf R}^n)\times L^1(T; {\bf R}^n)$ such that for some $U\subset T$ measurable and some $\{(\Gamma_0, \Lambda_0)\}\in{AC}(T; {\bf R}^n)\times L^\infty(T; {\bf R}^n)$, $(\Gamma_q(t), \Lambda_q(t))\to(\Gamma_0(t), \Lambda_0(t))$ uniformly on $U$, then for all $q\in{\bf N}$, $c(t, \Gamma_q(t), \Lambda_q(t))$ is measurable on $U$ and

Note that all conditions above concerning the functions $L$, $L_\gamma$ $(\gamma = 1, \ldots, K)$ and $\varphi$, are satisfied if the functions $L$, $L_\gamma$ $(\gamma = 1, \ldots, K)$, $\varphi$ and their first and second derivatives with respect to $x$ and ${\dot x}$ are continuous on $T \times {\bf R}^n \times {\bf R}^n$.

Set

We shall use the notation $x_b$ to denote any element $x_b: = (x, b)\in{\cal A}$. Let ${\cal B}$ any subset of ${\bf R}^p$ which we shall call the set of parameters. The parametric calculus of variations problem we shall deal with, denoted by (P), is that of minimizing the functional

over all $x_b\in{\cal A}$ satisfying the constraints

Elements $b = (b_1, \ldots, b_p)^\ast$ (the notation $\ast$ denotes transpose) in ${\cal B}$ will be called parameters, elements $x_b$ in ${\cal A}$ will be called arcs or trajectories, and a trajectory $x_b$ is admissible if it satisfies the constraints. The notation ${x_0}_{b_0}$ refers to an element $(x_0, b_0)\in{\cal A}$.

Let us now introduce some definitions which will be used throughout the paper.

${\bullet}$ An arc ${x_0}_{b_0}$ solves $(P)$ if it is admissible and $I({x_0}_{b_0})\le I(x_b)$ for all admissible arcs $x_b$. For strong local minima, an admissible arc ${x_0}_{b_0}$ is called a strong minimum of (P) if it is a minimum of $I$ relative to the following norm

that is, if for some $\epsilon > 0$, $I({x_0}_{b_0})\le I(x_b)$ for all admissible arcs satisfying $\|x_b-{x_0}_{b_0}\| < \epsilon$.

${\bullet}$ For all $x\in {\cal X}$, we use the notation $({\tilde x}(t))$ in order to represent $(t, x(t), {\dot x}(t))$. Also, $({\tilde x}_0(t))$ represents $(t, x_0(t), {\dot x}_0(t))$.

$\bullet$ Given $K$ real numbers $\lambda_1, \ldots, \lambda_K$, for any $x_b$ admissible define the functional $I_0$ by

where $l_0\colon{\bf R}^p\to{\bf R}$ is given by

and $L_0:\; T \times {\bf R}^n \times {\bf R}^n\to{\bf R}$ is given by

$\bullet$ Given $\lambda_1, \ldots, \lambda_K$, for all $(t, x, {\dot x}, \rho, \mu)\in T\times{\bf R}^n\times{\bf R}^n\times{\bf R}^n\times{\bf R}^s$, define the Hamiltonian of the problem by

where $\rho\in{\bf R}^n$ denotes the adjoint variable and $\mu\in{\bf R}^s$ is the associated multiplier of the mixed constraints.

$\bullet$ Given $(\rho, \mu)\in {\cal X}\times{\cal U}_s$, and $\lambda_1, \ldots, \lambda_K$, for all $(t, x, {\dot x}) \in T \times {\bf R}^n \times {\bf R}^n$, define the following function associated to the Hamiltonian,

$\bullet$ Given $(\rho, \mu)\in {\cal X}\times {\cal U}_s$ and $\lambda_1, \ldots, \lambda_K$, for any $x_b$ admissible define the functional $J_0$ by

$\bullet$ The notation $y_\beta$ refers to any element $(y, \beta)$ in ${\cal A}$.

$\bullet$ Given $(\rho, \mu)\in {\cal X}\times {\cal U}_s$, and $\lambda_1, \ldots, \lambda_K$, for any $x_b\in{\cal A}$ with ${\dot x}\in L^\infty(T; {\bf R}^n)$ and any $y_\beta\in {\cal A}$ consider the first variations of $J_0$ and $I_\gamma$ $(\gamma = 1, \ldots, K)$ with respect to $x_b$ over $y_\beta$ which are given, respectively, by

$\bullet$ For all $(t, x, {\dot x}) \in T \times {\bf R}^n \times {\bf R}^n$, denote by

the set of active indices of $(t, x, {\dot x})$ with respect to the mixed inequality constraints.

$\bullet$ For all $x_b\in{\cal A}$, denote by

the set of active indices of $x_b$ with respect to the isoperimetric inequality constraints.

$\bullet$ Given $x_b\in{\cal A}$, let ${\cal Y}(x_b)$ be the set of all $y_\beta\in {\cal A}$ with ${\dot y}\in L^2(T; {\bf R}^n)$ satisfying

The set ${\cal Y}(x_b)$ will be called the set of admissible variations along $x_b$.

$\bullet$ Given $(\rho, \mu)\in {\cal X}\times {\cal U}_s$, and $\lambda_1, \ldots, \lambda_K$, for any $x_b\in{\cal A}$ with ${\dot x}\in L^\infty(T; {\bf R}^n)$ and any $y_\beta\in {\cal A}$ with ${\dot y}\in L^2(T; {\bf R}^n)$, we define the second variation of $J_0$ with respect to $x_b$ over $y_\beta$, by

where for all $(t, y, {\dot y}) \in T \times {\bf R}^n \times {\bf R}^n$,

$\bullet$ Given $(\rho, \mu)\in {\cal X}\times {\cal U}_s$, $\lambda_1, \ldots, \lambda_K$ and ${x_0}_{b_0}\in {\cal A}$, we say that ${x_0}_{b_0}$ is singular, if for some $\tau\in T$, $|H_{{\dot x}{\dot x}}({\tilde x}_0(\tau), \rho(\tau), \mu(\tau))| = 0$. It satisfies the Legendre condition if

and the strengthened Legendre condition, if $F_{0{\dot x}{\dot x}}({\tilde x}_0(t)) > 0$ $(t\in T)$.

$\bullet$ Denote by $E_0$ the Weierstrass excess function of $F_0$, given by

$\bullet$ Similarly, the Weierstrass excess function of $L_\gamma$ $(\gamma = 1, \ldots, K)$, is given by

$\bullet$ For all $\pi = (\pi_1, \ldots, \pi_n)^\ast\in{\bf R}^n$, set

$\bullet$ For all $x\in {\cal X}$, define

$\bullet$ As we mentioned above the symbol $\ast$ denotes transpose.

It is well-known that, under certain normality assumptions (see for example ^[10]), if ${x_0}_{b_0}$ is a strong minimum of (P), then there exist $\rho\in{\cal X}$ and $\mu\in{\cal U}_s$ with $\mu_\alpha(t)\ge0$ and $\mu_\alpha(t)\varphi_\alpha({\tilde x}_0(t)) = 0$ $(\alpha\in R, \text{a.e. in}\ T)$ and multipliers $\lambda_1, \ldots, \lambda_K$ with $\lambda_i\ge0$ and $\lambda_i I_i({x_0}_{b_0}) = 0$ $(i = 1, \ldots, k)$ such that

The relations given above are the Euler-Lagrange equations of the constrained problem (P) and if $(x_0, \rho, \mu)$ satisfies the Euler-Lagrange equations, then $(x_0, \rho, \mu)$ will be called an extremal.

The following theorem is the main result of the article. Given an admissible arc ${x_0}_{b_0}$ with ${\dot x}_0$ neither continuous nor piecewise continuous but only essentially bounded, this theorem gives sufficient conditions assuring that ${x_0}_{b_0}$ is a strong minimum of problem (P). Hypothesis (ⅰ) of Theorem 2.1 is known as the transversality condition, hypothesis (ⅱ) arises from the original proof of the theorem, condition (ⅲ) is the necessary condition of Legendre, but not its strengthened version, hypothesis (ⅳ) is the positivity of the second variation over the set of all nonnull admissible variations and finally, hypothesis (ⅴ) involves three conditions related to the Weierstrass excess functions.

2.1 Theorem: Let ${x_0}_{b_0}$ be an admissible arc with ${\dot x}_0\in L^\infty(T; {\bf R}^n)$. Assume that ${\cal I}_a({\tilde x}_0(\cdot))$ is piecewise constant on $T$, and there exist $(\rho, \mu)\in {\cal X}\times {\cal U}_s$ with $\mu_\alpha(t)\ge0$ and $\mu_\alpha(t)\varphi_\alpha({\tilde x}_0(t)) = 0$ $(\alpha\in R, \text{a.e. in}\ T)$, two positive numbers $h, \epsilon$, and multipliers $\lambda_1, \ldots, \lambda_K$ with $\lambda_i\ge0$ and $\lambda_iI_i({x_0}_{b_0}) = 0$ $(i = 1, \ldots, k)$ such that $(x_0, \rho, \mu)$ is an extremal and the following holds:

$({\rm i})$ $l_0^\prime(b_0) + \rho^\ast(t_1)\Psi_1^\prime(b_0) - \rho^\ast(t_0)\Psi_0^\prime(b_0) = 0$.

$({\rm ii})$ $\rho^\ast(t_1)\Psi_1^{\prime\prime}(b_0;\beta) - \rho^\ast(t_0)\Psi_0^{\prime\prime}(b_0;\beta) \ge 0 \, \, for\ all \ \beta\in {\bf{R}}^p$.

$({\rm iii})$ $H_{{\dot x}{\dot x}}({\tilde x}_0(t), \rho(t), \mu(t))\le0$ $(\text{a.e. in}\ T)$.

$({\rm iv})$ $J^{\prime\prime}_0({x_0}_{b_0};y_\beta) > 0$ for all nonnull $y_\beta\in {\cal Y}({x_0}_{b_0})$.

$({\rm v})$ For all $x_b$ admissible with $\|x-x_0\|_C < \epsilon$,

${\rm \bf a.}$ $E_0(t, x(t), {\dot x}_0(t), {\dot x}(t)) \ge 0$ $(a.e.\ in\ T)$.

${\rm \bf b.}$ $\int_{t_0}^{t_1} E_0(t, x(t), {\dot x}_0(t), {\dot x}(t)) dt \ge h\int_{t_0}^{t_1} V({\dot x}(t)-{\dot x}_0(t))dt$.

${\rm \bf c.}$ $\int_{t_0}^{t_1} E_0(t, x(t), {\dot x}_0(t), {\dot x}(t))dt \ge h|\int_{t_0}^{t_1} E_\gamma(t, x(t), {\dot x}_0(t), {\dot x}(t))dt|$ $(\gamma = 1, \ldots, K)$.

Then for some $\theta_1, \theta_2 > 0$ and all admissible trajectories $x_b$ satisfying $\|x_b-{x_0}_{b_0}\| < \theta_1$,

In particular, ${x_0}_{b_0}$ is a strong minimum of (P).

3.   Statement of a nonparametric problem of Bolza and a fundamental result

Suppose we are given an interval $T: = [t_0, t_1]$ in ${\bf R}$, two sets ${\cal B}_0, {\cal B}_1\subset {\bf R}^n$ and functions $\ell(x_1, x_2)\colon{\bf R}^n\times{\bf R}^n\to{\bf R}$, $\ell_\gamma(x_1, x_2)\colon{\bf R}^n\times{\bf R}^n\to{\bf R}$ $(\gamma = 1, \ldots, K)$, ${\cal L}(t, x, {\dot x}):\; T \times {\bf R}^n \times {\bf R}^n\to{\bf R}$, ${\cal L}_\gamma(t, x, {\dot x}):\; T \times {\bf R}^n \times {\bf R}^n\to{\bf R}$ $(\gamma = 1, \ldots, K)$ and $\phi(t, x, {\dot x}):\; T \times {\bf R}^n \times {\bf R}^n\to{\bf R}^s$. Set

where $R: = \{1, \ldots, r\}$ and $S: = \{r+1, \ldots, s\}$ $(r = 0, 1, \ldots, s)$. If $r = 0$ then $R = \emptyset$ and we disregard statements involving $\varphi_\alpha$. Similarly, if $r = s$ then $S = \emptyset$ and we disregard statements involving $\varphi_\beta$.

It will be assumed throughout this section that ${\cal L}$, ${\cal L}_\gamma$ $(\gamma = 1, \ldots, K)$ and $\phi = (\phi_1, \ldots, \phi_s)$ have first and second derivatives with respect to $x$ and ${\dot x}$. Moreover, we shall assume that the functions $\ell$, $\ell_\gamma$ $(\gamma = 1, \ldots, K)$ are of class $C^2$ on ${\bf R}^n\times{\bf R}^n$.

Also, if we denote by $c(t, x, {\dot x})$ either ${\cal L}(t, x, {\dot x})$, ${\cal L}_\gamma(t, x, {\dot x})$ $(\gamma = 1, \ldots, K)$, $\phi(t, x, {\dot x})$ or any of its partial derivatives of order less than or equal to two with respect to $x$ and ${\dot x}$, we shall assume that all the assumptions posed in Section 2 in the statement of the problem are satisfied.

As in Section 2, ${\cal X}$ will denote the space of absolutely continuous functions mapping $T$ to ${\bf R}^n$ and ${\cal U}_s: = L^\infty(T; {\bf R}^s)$ the space of essentially bounded functions mapping $T$ to ${\bf R}^s$.

The nonparametric calculus of variations problem we shall deal with, denoted by $(\bar P)$, consists in minimizing the functional

over all $x\in{\cal X}$ satisfying the constraints

Elements $x$ in ${\cal X}$ will be called arcs or trajectories, and a trajectory $x$ is admissible if it satisfies the constraints.

An arc $x_0$ solves $(\bar P)$ if it is admissible and ${\cal J}(x_0)\le {\cal J}(x)$ for all admissible arcs $x$. For strong minima, an admissible arc $x_0$ is called a strong minimum of $(\bar P)$ if it is a minimum of ${\cal J}$ relative to the norm

that is, if for some $\epsilon > 0$, ${\cal J}(x_0)\le {\cal J}(x)$ for all admissible arcs satisfying $\|x-x_0\| < \epsilon$.

Let $\Psi\colon{\bf R}^n\to{\bf R}^n\times{\bf R}^n$ be any function of class $C^2$ such that ${\cal B}_0\times{\cal B}_1\subset\Psi({\bf R}^n)$. Associate the nonparametric problem $(\bar P)$ with the parametric problem of Section 2, which we denote by $(P_\Psi)$, that is, $(P_\Psi)$ will be the parametric problem given in Section 2, with $p = n$, ${\cal B} = \Psi^{-1}({\cal B}_0\times{\cal B}_1)$, $l = \ell\circ\Psi$, $l_\gamma = \ell_\gamma\circ\Psi$ $(\gamma = 1, \ldots, K)$, $L = {\cal L}$, $L_\gamma = {\cal L}_\gamma$ $(\gamma = 1, \ldots, K)$, $\varphi = \phi$ and $\Psi_0, \Psi_1$ the components of $\Psi$, that is, $\Psi = (\Psi_0, \Psi_1)$. Recall that the notation $x_b$ means $(x, b)$ where $b\in{\bf R}^n$ is a parameter.

3.1 Lemma: The following is satisfied:

${\rm (i)}$ $x_b$ is an admissible arc of $(P_\Psi)$ if and only if $x$ is an admissible arc of $(\bar P)$ and $b\in\Psi^{-1}(x(t_0), x(t_1))$.

${\rm (ii)}$ If $x_b$ is an admissible arc of $(P_\Psi)$, then

${\rm (iii)}$ If ${x_0}_{b_0}$ is a solution of $(P_\Psi)$, then $x_0$ is a solution of $(\bar P)$.

Proof: Conditions (ⅰ) and (ⅱ) follow from the definitions of the problems. Now, let $x$ an admissible arc of $(\bar P)$ and let $b\in\Psi^{-1}(x(t_0), x(t_1))$. By (ⅰ), $x_0$ is an admissible arc of $(\bar P)$ and $x_b$ is an admissible arc of $(P_\Psi)$. Then by (ⅱ),

which shows (ⅲ).

The following corollary which is a consequence of Theorem 2.1 and Lemma 3.1, provides a set of sufficient conditions of problem $(\bar P)$.

3.2 Corollary: Let $\Psi\colon{\bf R}^n\to{\bf R}^n\times{\bf R}^n$ be any function of class $C^2$ such that ${\cal B}_0\times{\cal B}_1\subset\Psi({\bf R}^n)$ and let $(P_\Psi)$ be the parametric problem defined in the previous paragraph of Lemma 3.1. Let ${x_0}_{b_0}$ be an admissible arc of $(P_\Psi)$ with ${\dot x}_0\in L^\infty(T; {\bf R}^n)$. Assume that ${\cal I}_a({\tilde x}_0(\cdot))$ is piecewise constant on $T$, and there exist $(\rho, \mu)\in {\cal X}\times {\cal U}_s$ with $\mu_\alpha(t)\ge0$ and $\mu_\alpha(t)\varphi_\alpha({\tilde x}_0(t)) = 0$ $(\alpha\in R, a.e. \ in\ T)$, two positive numbers $h, \epsilon$, and multipliers $\lambda_1, \ldots, \lambda_K$ with $\lambda_i\ge0$ and $\lambda_iI_i({x_0}_{b_0}) = 0$ $(i = 1, \ldots, k)$ such that $(x_0, \rho, \mu)$ is an extremal and the following holds:

$({\rm i})$ $l_0^\prime(b_0) + \rho^\ast(t_1)\Psi_1^\prime(b_0) - \rho^\ast(t_0)\Psi_0^\prime(b_0) = 0$.

$({\rm ii})$ $\rho^\ast(t_1)\Psi_1^{\prime\prime}(b_0;\beta) - \rho^\ast(t_0)\Psi_0^{\prime\prime}(b_0;\beta) \ge 0 \, \, for\ all\ \beta\in {\bf{R}}^n$.

$({\rm iii})$ $H_{{\dot x}{\dot x}}({\tilde x}_0(t), \rho(t), \mu(t))\le0$ $(\text{a.e. in}\ T)$.

$({\rm iv})$ $J^{\prime\prime}_0({x_0}_{b_0};y_\beta) > 0$ for all nonnull $y_\beta\in {\cal Y}({x_0}_{b_0})$.

$({\rm v})$ For all $x_b$ admissible with $\|x-x_0\| < \epsilon$,

${\rm \bf a.}$ $E_0(t, x(t), {\dot x}_0(t), {\dot x}(t)) \ge 0$ $(a.e.\ in\ T)$.

${\rm \bf b.}$ $\int_{t_0}^{t_1} E_0(t, x(t), {\dot x}_0(t), {\dot x}(t)) dt \ge h\int_{t_0}^{t_1} V({\dot x}(t)-{\dot x}_0(t))dt$.

${\rm \bf c.}$ $\int_{t_0}^{t_1} E_0(t, x(t), {\dot x}_0(t), {\dot x}(t))dt \ge h|\int_{t_0}^{t_1} E_\gamma(t, x(t), {\dot x}_0(t), {\dot x}(t))dt|$ $(\gamma = 1, \ldots, K)$.

Then, $x_0$ is a strong minimum of $(\bar P)$.

4.   Auxiliary results

In this section we state two auxiliary results which will be used to prove Theorem 2.1. The proof of these results will be given in Section 6. As before ${\cal X}$ denotes ${AC}(T; {\bf R}^n)$.

In the following two lemmas, we shall assume that we are given $x_0\in{\cal X}$ and $\{x^q\}$ a sequence in ${\cal X}$ such that

For all $q\in{\bf N}$ and $t\in T$, let

We say that ${\dot x}^q(t)\to {\dot x}_0(t)$ almost uniformly on $T$, if for any $\epsilon > 0$, there exists $U_\epsilon\subset T$ measurable with $m(U_\epsilon) < \epsilon$ such that ${\dot x}^q(t)\to {\dot x}_0(t)$ uniformly on $T\setminus U_\epsilon$.

4.1 Lemma: For some subsequence of $\{x^q\}$, again denoted by $\{x^q\}$, and some $y_0\in{\cal X}$ with ${\dot y}_0\in L^2(T; {\bf R}^n)$, ${\dot x}^q(t)\to {\dot x}_0(t)$ almost uniformly on $T$, $y_q(t)\to y_0(t)$ uniformly on $T$ and $\{{\dot y}_q\}$ converges weakly in $L^1(T; {\bf R}^n)$ to ${\dot y}_0$.

4.2 Lemma: Let $U\subset T$ measurable, $R_0\in L^\infty(U; {\bf R}^{n\times n})$ and $\{R_q\}$ a sequence in $L^\infty(U; {\bf R}^{n\times n})$. If ${\dot x}^q(t)\to{\dot x}_0(t)$ uniformly on $U$, $R_q(t)\to R_0(t)$ uniformly on $U$ and $R_0(t)\ge0$ $(t\in U)$, then

5.   Proof of Theorem 2.1

The proof of Theorem 2.1 will be divided in three Lemmas. In Lemmas 5.1, 5.2 and 5.3 below, we shall be assuming that all the hypotheses of Theorem 2.1 are satisfied. Before enunciating the lemmas, we shall introduce some definitions.

First of all, note that given $x = (x_1, \ldots, x_n)^\ast\in{\bf R}^n$ and $b = (b_1, \ldots, b_p)^\ast\in{\bf R}^p$, if we define $x{\bf i}, b{\bf j}\in{\bf R}^{n+p}$ by $x{\bf i}: = (x_1, \ldots, x_n, 0, \ldots, 0)^\ast$ and $b{\bf j}: = (0, \ldots, 0, b_1, \ldots, b_p)^\ast$, then

Define ${\tilde F}_0\colon T \times {\bf R}^{n+p} \times {\bf R}^n \to {\bf R}$ by

Observe that the Weierstrass excess function ${\tilde E}_0\colon T\times {\bf R}^{n+p} \times {\bf R}^n \times {\bf R}^n \to {\bf R}$ of ${\tilde F}_0$ is given by

It is clear that for all $(t, x, {\dot x}, u)\in T\times {\bf R}^n \times {\bf R}^n \times {\bf R}^n$ and all $b\in{\bf R}^p$,

Define

We have that $J_0(x_b) = {\tilde J}_0(x_b)$ for all $x_b\in{\cal A}$, and

where

and ${\tilde M}_0$, ${\tilde N}_0$ are given by

We have,

where

5.1 Lemma: For some $\nu, \kappa > 0$ $(\kappa\le\epsilon)$ and any admissible arc $x_b$ satisfying $\|x_b-{x_0}_{b_0}\| < \kappa$,

Proof: By condition (v)(b) of Theorem 2.1, given $x_b$ admissible with $\|x_b-{x_0}_{b_0}\| < \epsilon$,

On the other hand, by (2) and using $[t, b]$ in order to denote $(t, x(t){\bf i}+b{\bf j})$, observe that for some constants $c_0, c_1 > 0$, for all $x_b$ admissible with $\|x_b-{x_0}_{b_0}\| < 1$ and almost all $t\in T$,

Setting $\nu: = \max\{c_1, c_1(t_1-t_0)\}$, for all $x_b$ admissible with $\|x_b-{x_0}_{b_0}\| < 1$,

and so the conclusion of the lemma is obtained with $\kappa = \min\{\epsilon, 1\}.$

5.2 Lemma: If conclusion of Theorem 2.1 is false, then there exists a subsequence $\{x^q_{b_q}\}$ of admissible arcs such that

Proof: If conclusion of Theorem 2.1 is false, then for all $\theta_1, \theta_2 > 0$, there exists an admissible arc $x_b$ such that

Since

if $x_b$ is admissible, then $I(x_b)\ge J_0(x_b)$. Also, since

then $I({x_0}_{b_0}) = J_0({x_0}_{b_0})$. Therefore (3) implies that, for all $\theta_1, \theta_2 > 0$, there exists $x_b$ admissible with

Let $\kappa$ and $\nu$ be the positive numbers given in Lemma 5.1. Thus, if conclusion of Theorem 2.1 is false, then for all $q\in{\bf N}$, there exists $x_{b_q}^q$ admissible such that

Clearly, $D(x^q-x_0) = 0$ if and only if $x^q = x_0$. Then, by the second relation of (4),

Suppose $D(x^q-x_0) = 0$ for infinitely many $q$'s. For $i = 0, 1$, we have

Denoting by $(b_q, b_0)$ the line segment in ${\bf R}^p$ joining the points $b_q$ and $b_0$, by the second relation of (4), by condition (ⅰ) of Theorem 2.1, by (6), and the mean value theorem, there exists $\Xi_q\in(b_q, b_0)$ such that

Choose an appropriate subsequence of $\{(b_q-b_0)/|b_q-b_0|\}$ (without relabeling), such that

for some $\beta_0\in{\bf R}^p$ with $|\beta_0| = 1$. By (5),

For all $(t, \xi, {\dot x})\in T\times{\bf R}^{n+p}\times{\bf R}^n$ and $\gamma = 1, \ldots, K$, if we set

and for all $(t, \xi, {\dot x}, u)\in T\times{\bf R}^{n+p}\times{\bf R}^n\times{\bf R}^n$ and $\gamma = 1, \ldots, K$, if we set

we have that for all $x_b\in{\cal A}$ and $\gamma = 1, \ldots, K$,

where

and ${\tilde M}_\gamma$, ${\tilde N}_\gamma$ are given by

We have

where

Since ${x_0}_{b_q}$ and ${x_0}_{b_0}$ are admissible, for all $i\in i_a({x_0}_{b_0})$, we have

As one readily verifies, for all $\gamma = 1, \ldots, K$,

Then, for $i\in i_a({x_0}_{b_0})$,

On the other hand, once again since ${x_0}_{b_q}$ and ${x_0}_{b_0}$ are admissible, for all $j = k+1, \ldots, K$, we have

Then, for $j = k+1, \ldots, K$,

Consequently, by (9), (10) and (11), $0_{\beta_0}\in {\cal Y}({x_0}_{b_0})$.

By (7), (8) and condition (ⅱ) of Theorem 2.1, it follows that

which contradicts (ⅳ) of Theorem 2.1. Therefore, we may assume that for all $q\in{\bf N}$,

Since $(x_0, \rho, \mu)$ is an extremal, for all $q\in{\bf N}$,

and thus

By (1), the first relation of (4) and Lemma 5.1, for all $q\in{\bf N}$,

then, by (4), for all $q\in{\bf N}$,

By (12),

5.3 Lemma: If conclusion of Theorem 2.1 is false, then condition (iv) of Theorem 2.1 is false.

Proof: Let $\{x^q_{b_q}\}$ be the sequence of admissible arcs given in Lemma 5.2. Then,

Case (1): Suppose first that the sequence $\{(b_q-b_0)/d_q\}$ is bounded in ${\bf R}^p$.

For all $q\in{\bf N}$ and $t\in T$, define

By Lemma 4.1, for some $y_0\in{\cal X}$ with ${\dot y}_0\in L^2(T; {\bf R}^n)$ and a subsequence of $\{x^q\}$, again denoted by $\{x^q\}$, $\{{\dot y}_q\}$ converges weakly in $L^1(T; {\bf R}^n)$ to ${\dot y}_0$. Once again, by Lemma 4.1,

Since the sequence $\{(b_q-b_0)/d_q\}$ is bounded in ${\bf R}^p$, then we may assume that there exists some $\beta_0\in{\bf R}^p$ such that

First, we are going to show that for $i = 0, 1$,

Note first that for $i = 0, 1$ and all $q\in{\bf N}$, we have that

By (13), (14) and (16), we obtain (15). Now, we claim that

To prove it, observe that by (2), (13) and (14),

both uniformly on $T$. With this in mind and since $\{{\dot y}_q\}$ converges weakly in $L^1(T; {\bf R}^n)$ to ${\dot y}_0$, we have

Since $(x_0, \rho, \mu)$ is an extremal and by condition (ⅰ) of Theorem 2.1, it follows that

Consequently, by (1), (4), (19), and condition (ⅱ) of Theorem 2.1,

Now, let us show that

To this end, let $U$ a measurable subset of $T$ such that ${\dot x}^q(t)\to {\dot x}_0(t)$ uniformly on $U$. For all $q\in{\bf N}$ and $t\in U$, we have that

where

Clearly,

By condition (ⅲ) of Theorem 2.1, we have

With this in mind, and since by (v)(a) of Theorem 2.1 for all $q\in{\bf N}$,

by Lemma 4.2,

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

belongs to $L^1(T; {\bf R})$, this inequality holds when $U = T$, and this establishes (21). With this in mind, by (18) and (20), we have

Now, let us show that ${y_0}_{\beta_0}\not\equiv(0, 0)$. By (20), the first conclusion of Lemma 5.1, the fact that $V(\pi)\le |\pi|^2/2$ for all $\pi\in{\bf R}^n$,

With this in mind and (18), the fact that ${y_0}_{\beta_0}\equiv(0, 0)$ contradicts the positivity of $h$ and this establishes (17).

Let us now show that

To this end, note that, for all $\gamma = 0, 1, \ldots, K$,

all uniformly on $T$ and $\{{\dot y}_q\}$ converges weakly in $L^1(T; {\bf R}^n)$ to ${\dot y}_0$, then for all $\gamma = 0, 1, \ldots, K$,

As in (19), we have

By (4), (23) and (24),

Since ${\tilde {\cal E}}_0({x_0}_{b_0};x^q_{b_q})\ge0$ $(q\in{\bf N})$, then

Therefore, by condition (v)(c) of Theorem 2.1, for all $\gamma = 1, \ldots, K$,

As for all $q\in{\bf N}$ and $i\in i_a({x_0}_{b_0})$,

then, by (23) and (25), for all $i\in i_a({x_0}_{b_0})$,

Therefore, since $y_q(t)\to y_0(t)$ uniformly on $T$, $\{{\dot y}_q\}$ converges weakly in $L^1(T; {\bf R}^n)$ to ${\dot y}_0$ and $(b_q-b_0)/d_q\to\beta_0$, then for all $i\in i_a({x_0}_{b_0})$,

which establishes (22).

Now, let us show that

Indeed, as for all $q\in{\bf N}$ and $j = k+1, \ldots, K$,

by (23) and (25), for all $j = k+1, \ldots, K$,

which is precisely (26).

We claim that, for all $\alpha\in {\cal I}_a({\tilde x}_0(t))$,

Indeed, for all $\alpha\in R$, $q\in{\bf N}$, almost all $t\in T$ and $\lambda\in[0, 1]$, define

where as usual $({\tilde x}^q(t)): = (t, x^q(t), {\dot x}^q(t))$ $(\text{a.e. in}\ T)$. Given $t\in[t_0, t_1)$ a point of continuity of ${\cal I}_a({\tilde x}_0(\cdot))$ and $\alpha\in {\cal I}_a({\tilde x}_0(t))$, since ${\cal I}_a({\tilde x}_0(\cdot))$ is piecewise constant on $T$, there exists an interval $[t, \bar t]\subset T$ with $t < \bar t$ such that $\varphi_\alpha({\tilde x}_0(\tau)) = 0$ for all $\tau\in[t, \bar t]$. Using the notation

we have

Since $U$ can be chosen to differ from $T$ by a set of an arbitrarily small measure, then

If $Z_0^\alpha(\tau) < 0$ on a measurable set $\Theta$ such that $\Theta\subset[t, \bar t]$ and $m(\Theta) > 0$, then

which is a contradiction. Therefore, $Z_0^\alpha(\tau)\ge0$ $\text{a.e. in}\ [t, \bar t]$ with $t\in[t_0, t_1)$ an arbitrary point of continuity of ${\cal I}_a({\tilde x}_0(\cdot))$ and, therefore, $Z_0^\alpha(t)\ge0$ for almost all $t\in T$ which shows (27).

Now, let us show that for all $\beta\in S$,

Indeed, for all $\beta\in S$, $q\in{\bf N}$, almost all $t\in T$ and $\lambda\in[0, 1]$, define

For all $\beta\in S$, $q\in{\bf N}$ and almost all $t\in T$, we have

Therefore, for all $\beta\in S$, $q\in{\bf N}$ and almost all $t\in T$,

By (29), for all $t\in T$ and $\beta\in S$,

Since, as before, $U$ can be chosen to differ from $T$ by a set of an arbitrarily small measure, then for all $t\in T$ and $\beta\in S$,

and so (28) is verified. Consequently, from (15), (22), (26), (27) and (28), ${y_0}_{\beta_0}\in {\cal Y}({x_0}_{b_0})$. This fact together with (17) contradicts condition (ⅳ) of Theorem 2.1.

Case (2): Now, suppose that the sequence $\{(b_q-b_0)/d_q\}$ is not bounded. Then,

Choose an appropriate subsequence of $\{(b_q-b_0)/|b_q-b_0|\}$ (without relabeling), and $\bar\beta_0\in{\bf R}^p$ with $|\bar\beta_0| = 1$, such that

For all $q\in{\bf N}$ and $t\in T$, define

By Lemma 4.1 and (30),

For $i = 0, 1$ and all $q\in{\bf N}$, we have

By (31), (32) and (33), for $i = 0, 1$,

Now, by (2), (31) and (32),

both uniformly on $T$. Together with Lemma 4.1, this implies that

As in (19), we have

Even more, by (1), (4), (36), and condition (ⅱ) of Theorem 2.1,

Consequently, since ${\tilde {\cal E}}_0({x_0}_{b_0};x^q_{b_q})\ge0$ $(q\in{\bf N})$, by (35) and (37),

Let us now show that for all $i\in i_a({x_0}_{b_0})$,

To prove it, note that since

and $\{{\dot y}_q\}$ converges weakly in $L^1(T; {\bf R}^n)$ to ${\dot y}_0$, for all $\gamma = 0, 1, \ldots, K$,

As in (24),

Then, by (4), (40) and (41),

Thus, by condition (v)(c) of Theorem 2.1, for all $\gamma = 1, \ldots, K$,

As for all $q\in{\bf N}$ and $i\in i_a({x_0}_{b_0})$,

then, by (40) and (42), for all $i\in i_a({x_0}_{b_0})$,

Hence, since

for all $i\in i_a({x_0}_{b_0})$,

which establishes (39). Finally, let us show that for all $j = k+1, \ldots, K$,

Indeed, as for all $q\in{\bf N}$ and $j = k+1, \ldots, K$,

then, by (40) and (42), for all $j = k+1, \ldots, K$,

which is precisely (43). Consequently, (34), (38), (39) and (43) contradicts condition (ⅳ) of Theorem 2.1 and this completes the proof of Theorem 2.1.

6.   Proof of Lemmas 4.1 and 4.2

Proof of Lemma 4.1: For all $q\in{\bf N}$ and almost all $t\in T$, define

For all $q\in{\bf N}$, note that

Clearly $\lim_{q\to\infty}c_q = 1$. Then, there exist some subsequence of $\{x^q\}$, again denoted by $\{x^q\}$, some $\bar y_0\in{\bf R}^n$ and some $\sigma_0\in L^2(T; {\bf R}^n)$ such that

As $W_q^2(t) \ge W_q(t) \ge 1$ for all $q\in{\bf N}$ and for almost all $t\in T$, we have

Thus, it follows that

Note also that

Then for any $h\in L^\infty(T; {\bf R}^n)$,

and so

Therefore, $\{{\dot y}_q\}$ converges weakly in $L^1(T; {\bf R}^n)$ to $\sigma_0$. Hence, $\{{\dot y}_q\}$ is equi-integrable on $T$ and therefore the sequence $\{y_q\}$ is equi-continuous on $T$. Thus, if $y_0(t): = \bar y_0 + \int_{t_0}^t\sigma_0(\tau)d\tau$, then, $y_0\in{\cal X}$ with ${\dot y}_0\in L^2(T; {\bf R}^n)$ and

Now, let us show that ${\dot x}^q(t)\to {\dot x}_0(t)$ almost uniformly on $T$. For almost all $t\in T$, define

Observe that

From these relations, we have

Consequently, $\|{\dot x}^q-{\dot x}_0\|_1\to0$ and so some subsequence of $\{{\dot x}^q\}$ converges almost uniformly to ${\dot x}_0$ on $T$.

Proof of Lemma 4.2: Recall the definition of $W_q$ given in the proof of Lemma 4.1. As $W_q(t)\to1$ uniformly on $U$, then for all $h\in L^2(U; {\bf R}^n)$,

that is, $\{{\dot y}_q\}$ converges weakly in $L^2(U; {\bf R}^n)$ to ${\dot y}_0$. As $R_0(t)\ge0$ $(t\in U)$, the function

is convex in $L^2(U; {\bf R}^n)$ and since this function is strongly continuous on $L^2(U; {\bf R}^n)$, then this function is weakly lower semicontinuous in $L^2(U; {\bf R}^n)$. Thus,

Since $R_q(t)\to R_0(t)$ uniformly on $U$, it follows that

7.   Example

In this section, we show with an example how our sufficiency theory is able to detect optimality even when the proposed extremal to be a strong minimum is singular and its derivative is only essentially bounded. It is worthwhile observing that the initial and final end-points of the states of admissible trajectories are not restricted to belong to any manifold described by any smooth function, in contrast, these boundary points must only lie in the set of real numbers, that is, these boundary points are completely free.

In Example 7.1 since no isoperimetric constraints occur $l_0$, $L_0$, $F_0$, $E_0$ and $J^{\prime\prime}_0$ correspond simply to $l$, $L$, $F$, $E$ and $J^{\prime\prime}$ respectively.

7.1 Example: Let $x_0\colon[0, 1]\to{\bf R}$ be any absolutely continuous function with ${\dot x}_0\in L^\infty([0, 1];{\bf R})$ and $x_0(0) = x_0(1) = 0$. Consider the nonparametric problem $(\bar P)$ of minimizing

over all arcs $x\in{\cal X}$ satisfying the constraints

where

${\cal X}: = {AC}([0, 1];{\bf R})$ and $c(t, x, {\dot x})$ denotes either

$\phi_1(t, x, {\dot x})$, or any of its partial derivatives of order less than or equal to two with respect to $x$ and ${\dot x}$.

For this problem we shall consider the data of the nonparametric problem given in Section 3 which are given by $T = [0, 1]$, $n = 1$, $r = 1$, $s = 1$, $k = K = 0$, ${\cal B}_0 = {\bf R}$, ${\cal B}_1 = {\bf R}$, $\ell(x_1, x_2) = x_1^2-x_2$, ${\cal L}(t, x, {\dot x}) = \exp(t({\dot x}-{\dot x}_0(t)))+x$, $\phi_1(t, x, {\dot x}) = ({\dot x}-{\dot x}_0(t))^2-\exp(t({\dot x}-{\dot x}_0(t)))+t({\dot x}-{\dot x}_0(t))+1$.

Clearly, all the assumptions posed in the statement of the problem are easily verified.

Also, it is evident that the trajectory $x_0$ is admissible of $(\bar P)$. Let $\Psi\colon{\bf R}\to{\bf R}\times{\bf R}$ be defined by $\Psi(b): = (b, b)$. Clearly, $\Psi$ is $C^2$ in ${\bf R}$ and ${\cal B}_0\times{\cal B}_1\subset\Psi({\bf R})$. The associated parametric problem of Section 2 denoted by $(P_\Psi)$ has the following data, $p = 1$, ${\cal B} = \Psi^{-1}({\cal B}_0\times{\cal B}_1) = {\bf R}$, $l = \ell\circ\Psi$, $L = {\cal L}$, $\varphi = \varphi_1 = \phi_1$ and $\Psi_0$, $\Psi_1$ the components of $\Psi$, that is, $\Psi_0(b) = b$, $\Psi_1(b) = b$ $(b\in{\bf R})$. Recall that the notation $x_b$ means $(x, b)$ where $b\in{\bf R}$ is a parameter.

Observe that if we set $b_0: = 0$, then ${x_0}_{b_0}$ is admissible of $(P_\Psi)$. Also, clearly ${\cal I}_a({\tilde x}_0(\cdot))$ is constant on $T$. Let $(\rho(t), \mu_1(t)): = (t, 0)$ $(t\in T)$ and note that $(\rho, \mu)\in{\cal X}\times{\cal U}_1$, $\mu_1(t)\ge0$, $\mu_1(t)\varphi_1({\tilde x}_0(t)) = 0$ $(\text{a.e. in}\ T)$.

Now, observe that the Hamiltonian $H$ is given by

Also, note that

and so $(x_0, \rho, \mu)$ satisfies the Euler-Lagrange equations which are given by

Thus, $(x_0, \rho, \mu)$ is an extremal. As $\Psi_0(b) = b$, $\Psi_1(b) = b$, $l(b) = b^2-b$ $(b\in{\bf R})$, then

and so condition (ⅰ) of Corollary 3.2 is satisfied. Also, as one readily verifies,

and so condition (ⅱ) of Corollary 3.2 is fulfilled.

Additionally, as

it follows that (ⅲ) of Corollary 3.2 is satisfied and $(x_0, \rho, \mu)$ is singular since

As $\varphi_{1x}({\tilde x}_0(t)) = \varphi_{1{\dot x}}({\tilde x}_0(t)) = 0$ $(\text{a.e. in}\ T)$, then ${\cal Y}({x_0}_{b_0})$ is given by all $y_\beta\in{\cal X}\times{\bf R}$ with ${\dot y}\in L^2(T; {\bf R})$ satisfying $y(0) = \beta$, $y(1) = \beta$. We have,

for all $y_\beta\in{\cal Y}({x_0}_{b_0})$, $y_\beta\not = (0, 0)$, and hence (ⅳ) of Corollary 3.2 is satisfied. Now, observe that for all $(t, x, {\dot x})\in T\times{\bf R}\times{\bf R}$,

Hence for almost all $t\in T$, if $x_b$ is admissible,

which implies that (v)(a) of Corollary 3.2 is verified with any $\epsilon > 0$. Finally, for all $x_b$ admissible, we have

Therefore, (v)(b) of Corollary 3.2 is satisfied with any $\epsilon > 0$ and $h = 2$. Since $k = K = 0$, it is evident that (v)(c) of Corollary 3.2 is also verified with any $\epsilon > 0$ and $h = 2$. By Corollary 3.2, $x_0$ is a strong minimum of $(\bar P)$.

Acknowledgment

The author is grateful to Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México, for the support given by the project PAPIIT-IN102220.

Conflict of interest

The author declares no conflict of interest.