Calculus of variations and optimal control; optimization Definition

Optimal control

Builds on Admissible control Control system Continuous function

Definition

Let $x' = f(x, u)$ be a control system on $\R^n$ with control set $U \subseteq \R^m$ . Fix a time horizon $T > 0$ , an initial state $x_0 \in \R^n$ and a closed target set $S \subseteq \R^n$ . Let $L \colon \R^n \times U \to \R$ and $\varphi \colon \R^n \to \R$ be continuous functions, the running cost and the terminal cost. For an admissible control $u$ for $(x_0, T, S)$ with trajectory $x_u$ , the cost is $J(u) = \varphi\big(x_u(T)\big) + \int_0^T L\big(x_u(t), u(t)\big) \, dt.$ An admissible control $u^*$ is an optimal control if $J(u^*) \le J(u)$ for every admissible control $u$ for $(x_0, T, S)$ . The pair $(u^*, x_{u^*})$ is then an optimal pair, and $x_{u^*}$ is an optimal trajectory.

Remarks

The integral exists, because $x_u$ is continuous and $u$ is essentially bounded, so the integrand is a bounded measurable function. A cost with both terms is in Bolza form; $L = 0$ gives the Mayer form and $\varphi = 0$ the Lagrange form. An optimal control need not exist, and when it exists it need not be unique.

Example: $x' = u$ on $\R$ with $U = [-1, 1]$ , $x_0 = 0$ , $T = 1$ , $S = \R$ , $L = 0$ and $\varphi(x) = -x$ . Every admissible $u$ gives $x_u(1) = \int_0^1 u(t) \, dt \le 1$ , so $J(u) \ge -1$ . The control $u^* = 1$ gives $J(u^*) = -1$ , so it is optimal.

Non-example: the control $u = 0$ in the same problem is admissible but not optimal, since $J(0) = 0 > -1 = J(u^*)$ .

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Tag	`EZ8NBZ8D0E`
URI	`m2://EZ8NBZ8D0E`
Kind	definition
Writing	W0 stub
MSC	49J15, 49K15, 93C15
Defines	Optimal control
Uses	Admissible control, Control system, Continuous function

Optimal control

Definition

Remarks

Reading path

Further reading

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