Optimal control of a stochastic system with an exponential-of-integral performance criterion
The author considers the nonlinear system described by the Itô stochastic differential equation: \[ dx(t)= h(x,u)dt+ \sigma(x) dw(t) \tag{1} \] where \(x\in \mathbb{R}^ n\) is the state, \(u\in \mathbb{R}^ m\) is the control and \(w(t)\) is an \(r\)-dimensional standard Wiener process. The cost functional (to be minimized) is the infinite-horizon or long-run average exponential-of-integral performance criterion: \[ J(u)= {1\over\theta} \lim_{T\to \infty} {1\over T}\ln E_ x^ u \Biggl[ \exp \biggl\{ \theta \int_ 0^ T c(x,u) dt \biggr\} \Biggr], \tag{2} \] where \(c(x,u)\) is a nonnegative continuous function and \(\theta\) is a scalar; \(E_ x^ u [\cdot ]\) denotes the expectation associated with the closed-loop system with control \(u\in U\) (admissible control). The author derives existence conditions for the state-feedback admissible controllers and optimality conditions. He discusses connections between the exponential-of-integral optimal control problem and stochastic differential games. The obtained results are illustrated by examples.
scientific article; zbMATH DE number 612406
scientific article