an:04058277
Zbl 0648.49003
Krasovskij, A. A.
Generalized solution of the control optimization problem with a nonclassical functional
EN
Sov. Phys., Dokl. 30, No. 10, 828-829 (1985); translation from Dokl. Akad. Nauk SSSR 284, No. 4, 808-811 (1985).
00181236
1985
j
49J15 49M99 93C15 49J55 93E20
nonclassical optimal control; generalized solution; Lyapunov equation
The paper deals with the control optimization problem for a dynamical process
\[
(*)\quad \dot x=f(x,t)+\phi (x,t)u,\quad x\in E\quad n,\quad u\in E\quad r
\]
with the optimal criterion
\[
I=\int^{t_ 2}_{t_ 1}\{Q_ g(x,\theta)+\sum^{r}_{j=1}k_ j^{-1}[u\quad q_ j(\quad \theta)+p^{-1}u\quad q_{j opt}(\theta)]\}d\theta,
\]
where \(u_{opt}\) is an unknown optimal control minimizing I on the solution of (*), \(k_ j>0\), \(p^{-1}+q^{-1}=1\), \(u\) \(q_ j\) is an even function of \(u_ j\). It is shown that the functional I has a unique minimum for \(u=u_{opt}\) defined by
\[
(\partial V/\partial x)\phi (x,t)=-(\partial /\partial u)U_ g(u_{opt},t),
\]
where \(V=V(x,t)\) is the solution of the Lyapunov equation for the uncontrolled process (*) (u\(\equiv 0)\). Moreover, the paper shows that the optimization of stochastic control with uncertainty concerning the observable noise leads to functionals in the form of conditional mathematical expectations which are used for control processes optimized in a deterministic way. The optimal (or suboptimal) control in both the deterministic and the stochastic case can be created via the solution of the Lyapunov equation under adquate boundary conditions. An algorithm is presented that generates this solution.
W.Hejmo