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Existence theorems concerning simple integrals of the calculus of variations for discontinuous solutions. (English) Zbl 0618.49004

L. Cesari’s well-known book ”Optimization – theory and applications” [Appl. Math. 17 (1983; Zbl 0506.49001)] does not cover existence of discontinuous optimal trajectories in optimal control problems but in the bibliographical notes on p. 452 of that volume the author states that such problems would be discussed elsewhere.
In the present paper Cesari and his collaborators make good on that promise and applying direct methods of the calculus of variations prove existence of discontinuous solutions to certain one-dimensional variational problems with constraints on the derivative of the solutions which are chosen from a class of vector-valued functions defined on a compact interval \(I\subset {\mathbb{R}}\) such that some component functions are of bounded variation on \(I\) in the sense of Cesari (briefly, BVC\((I;{\mathbb{R}}))\) whereas all other component functions belong to the class ACG\((I;{\mathbb{R}}))\) of \(L^ 1(I;{\mathbb{R}})\)-functions which are a.e. on \(I\) equal to an absolutely continuous function. More specifically, the authors’ main existence theorem is as follows:
Let \(\alpha\), \(n\in {\mathbb{N}}\), \(1\leq \alpha \leq n-1\), \(t_ 1,t_ 2\in {\mathbb{R}}\), \(I:=[t_ 1,t_ 2]\), and \(A\subset {\mathbb{R}}^ 1\times {\mathbb{R}}^{n-1}\) be a compact set such that its projection onto the \({\mathbb{R}}^ 1\)-axis contains \(I\); furthermore let \(Q: A\to {\mathbb{R}}^ n\) be a set-valued mapping, \(M:=\{(t,x;\zeta)\in {\mathbb{R}}^{n+1}\times {\mathbb{R}}^ n\mid \zeta \in Q(t,x)\), \((t,x)\in A\}\) be a closed set and \(f: M\to {\mathbb{R}}\) be a lower semi-continuous function satisfying a superlinear Tonelli-Nagumo-type growth condition; the sets \(\tilde Q(t,x) := \{(z,\zeta)\in {\mathbb{R}}\times Q(t,x)\mid z\geq f(t,x;\zeta)\}\), \((t,x)\in A\) are assumed to be closed convex sets having at every \((t,x)\in A\) Cesari’s “property (Q)” (this in particular implies that \(f(t,x,\zeta)\) is a convex function with respect to the variable \(\zeta)\); moreover let \(C\) denote the class of all functions \(x=(y,z): I\to {\mathbb{R}}^{\alpha}\times {\mathbb{R}}^{n-\alpha}\) with \(y\in \text{ACG}(I;{\mathbb{R}}^{\alpha})\), \(z\in \text{BVC}(I;{\mathbb{R}}^{n-\alpha})\) and \((t,x(t))\in A\), \(t\in {\mathbb{R}}\), \(x'(t)\in Q(t,x(t))\) a.e. on \(I\) such that \(f(\cdot,x(\cdot),x'(\cdot))\) is \(L^ 1(I;{\mathbb{R}})\); let \(\Omega\subset C\) be a non-void subclass that is closed in the following sense: If \(x=(x,z)\in C\) and if there is a sequence \((x_ k)_{k\in {\mathbb{N}}}\subset \text{ACG}(I;{\mathbb{R}}^ n)\cap \Omega\) with the properties:
i) \(x_ k=(y_ k,z_ k)\), \(y_ k\subseteq \text{ACG}(I;{\mathbb{R}}^{\alpha})\), \(z_ k\in \text{ACG}(I;{\mathbb{R}}^{n-\alpha})\), \(k\in {\mathbb{N}},\)
ii) \((y_ k)_{k\in {\mathbb{N}}}\) converges uniformly on \(I\) to \(y\) and \((z_ k)_{k\in {\mathbb{N}}}\) converges pointwise a.e. on \(I\) to \(z\),
then \(x\) is an element of \(\Omega\). It is further assumed that there exists a constant \(L\in {\mathbb{R}}\), \(L>0\), with \(V^*(z)\leq L\) for all \(x=(y,z)\) from \(\Omega \cap \text{ACG}(I;{\mathbb{R}}^ n)\) where \(V^*(z)\) is the generalized variation of \(z\). Finally, for any \(x\in \Omega\) let \(\Gamma(x)\) denote the class of sequences \((x_ k)_{k\in {\mathbb{N}}}\subset \Omega \cap \text{ACG}(I;{\mathbb{R}}^ n)\) satisfying i), ii) above and let the extended real-valued functionals \(J^*, J: \Omega\to {\overline {\mathbb{R}}}\) be defined by \[ J(x):=\int^{t_ 2}_{t_ 1}f(t,x(t),x'(t))\,dt,\quad J^*(x):=\inf \{\lim_{\overline{k\to \infty}}J(x_ k)| (x_ k)_{k\in {\mathbb{N}}}\in \Gamma (x)\}, \] \(x\in \Omega\) (note that \(J^*\) is defined somewhat analogously to Lebesgue’s definition of surface area for nonparametric discontinuous surfaces).
Under the hypotheses above the authors prove existence of \(x\in \Omega\) with \[ J^*(x)=\min_{u\in \Omega}J^*(u)=\inf \{J(w)\mid w\in \Omega \cap \text{ACG}(I,{\mathbb{R}}^ n)\}; \] the proof is based on a lower semi-continuity and lower closure theorem for \(J\) and \(J^*.\)
In the case \(\alpha =0\) (excluded above) a corresponding existence theorem for a minimizer of \(J^*\) can be proved if instead of a Nagumo- Tonelli growth condition f satisfies the inequality \(f(t,x,\zeta)\geq g(t)\) for all \((t,x,\zeta)\in M\) with a given \(L^ 1(I;{\mathbb{R}})\)-function \(g\); the case \(\alpha =n\) (also excluded above) is essentially already dealt with in theorems 11.1.i), 11.1.ii) of Cesari’s book (loc. cit.). As a corollary to the above mentioned existence theorem the authors also prove existence of an optimal trajectory in related optimal control problems.
Reviewer: H. Böttger

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
93B05 Controllability
49Q99 Manifolds and measure-geometric topics
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)

Citations:

Zbl 0506.49001
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References:

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