Calculus of variations and optimal control; optimization Optimal control: formulation and existence Problem

Bolza problem

Builds on Optimal control Control system Admissible control Functional Mayer and Lagrange problems Filippov's existence theorem Pontryagin's maximum principle

Problem

Let $U \subseteq \R^m$ be a nonempty set, and let $t_0 < t_1$ be fixed times. Let $f \colon [t_0, t_1] \times \R^n \times U \to \R^n$ and $L \colon [t_0, t_1] \times \R^n \times U \to \R$ be continuous. Let $\varphi \colon \R^n \times \R^n \to \R$ be continuous, and let $S \subseteq \R^n \times \R^n$ be a closed set of allowed endpoint pairs. An admissible control is a measurable map $u \colon [t_0, t_1] \to U$ whose trajectory $x$ of the control system $x'(t) = f(t, x(t), u(t))$ is absolutely continuous on $[t_0, t_1]$ and satisfies $(x(t_0), x(t_1)) \in S$ . The Bolza problem asks to minimize the functional $J(u) = \varphi(x(t_0), x(t_1)) + \int_{t_0}^{t_1} L(t, x(t), u(t)) \, dt$ over all admissible controls $u$ .

Remarks

The cost has two parts: an endpoint term $\varphi$ and a running term given by the integral of $L$ . The case $L = 0$ is the Mayer problem, and the case $\varphi = 0$ is the Lagrange problem; see Mayer and Lagrange problems. The three forms are equivalent: adding the state $y' = L(t, x, u)$ with $y(t_0) = 0$ turns the Bolza cost into the endpoint cost $\varphi(x(t_0), x(t_1)) + y(t_1)$ . In the classical calculus of variations the control is the velocity, $f(t, x, u) = u$ , and the problem is to minimize $\varphi(x(t_0), x(t_1)) + \int_{t_0}^{t_1} L(t, x, x') \, dt$ with possible side conditions. Necessary conditions for a minimizer are given by Pontryagin's maximum principle, with transversality conditions coming from $\varphi$ and $S$ . Existence of a minimizer needs extra hypotheses, such as compactness of $U$ and convexity of the set of velocities and costs; see Filippov's existence theorem.

Tag	`8Z55NXZWD5`
URI	`m2://8Z55NXZWD5`
Kind	problem
Writing	W0 stub
MSC	49K15, 49J15
Status	This copy of the site has no build data. Run `m2 status 8Z55NXZWD5`.
About	Optimal control
Uses	Optimal control, Control system, Admissible control, Functional, Mayer and Lagrange problems, Filippov's existence theorem, Pontryagin's maximum principle

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