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Stochastic hybrid control. (English) Zbl 0967.93097
The authors consider a complicated version of controlled stochastic systems. The time $t$ is measured continuously. The state of the system is represented by a continuous variable $x$ and a discrete variable $n$. Also, the control has two parts, a continuous type control $v$ that is a measurable stochastic process and a discrete-type (or impulse) control $k$ that is a sequence of random variables. The key point is the set-interface $D$ of which only the boundary is really used. Minimal and maximal set-interfaces are considered. When the state reaches the minimal set, a mandatory impulse (jump or switch) takes place, while if the state belongs to a maximal set, an optional impulse (jump or switch) may be applied, upon decision of the controller. Switching and jumps can be autonomous or totally controlled. A discounted marginal cost of the form $f\left(x\left(t\right),n\left(t\right),v\left(t\right)\right)exp\left(-{\int }_{0}^{t}c\left(x\left(s\right),n\left(s\right),v\left(s\right)\right)ds\right)$ is introduced and a control problem consists in its minimization. The authors demonstrate that the dynamic programming approach leads to some involved quasi-variational inequality. If the system is non-degenerate then the classic treatment can be used for the solution of the control problem, otherwise, a way is to use the so-called viscosity solutions that are described in the last part of the paper.
##### MSC:
 93E20 Optimal stochastic control (systems) 93B12 Variable structure systems 49L25 Viscosity solutions (infinite-dimensional problems)