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Approximating the linear quadratic optimal control law for hereditary systems with delays in the control. (English) Zbl 0651.93025

For the problem \[ \dot x(t)=\int^{0}_{-r}d_{\theta}\eta (t,\theta)x(t+\theta)+\sum^{k}_{i=0}B_ i(t)u(t-r_ i)+\int^{t}_{t-r}B(t,\theta)u(\theta)d\theta,\quad x(t)=\phi (t),\quad t\in [-r,0], \]
\[ (1)\quad \int^{T}_{- r}<x(s),Q(s)x(s)>d\mu +\int^{T}_{-r}| u(s)|^ 2ds\to \min \] an appropriate perturbed problem is considered where the functional (1) is approximated by \[ \int^{T}_{- r}<x(s),Q(s)x(s)>d\mu_{\pi}(s)+\int^{T}_{-r}| u(s)|^ 2ds,\quad \mu_{\pi}\to \mu. \] In the case \(\mu =\lambda +\delta_ T\), where \(\lambda\) denotes the Lebesgue measure and \(\delta_ T\) is the Dirac measure with support on \(\{\) \(T\}\), convergence of optimal controls and optimal trajectories is shown. For \(\mu =\sum^{n}_{i=1}a_ i\delta_{s_ i}\), \(s_ i\in [-r,T]\), \(S_ n=T\), the performance of an optimal feedback kernel is studied on the basis of which an algorithm is proposed. A numerical example is considered.
Reviewer: T.Tadumadze

MSC:

93B40 Computational methods in systems theory (MSC2010)
34K35 Control problems for functional-differential equations
93C05 Linear systems in control theory
49M99 Numerical methods in optimal control
65K10 Numerical optimization and variational techniques
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