An optimal control problem for stationary delay systems (Q5926578)

The author studies a vector system with delay \[ dx/dt= Ax(t)+ A_1x(t- h)+ bu(t). \] Here, the time \(t\) is restricted to an interval \([0,t_1]\), \(u(t)\) is a scalar piecewise continuous control, \(h>\theta> 0\) is the delay. The initial condition is specified: \(x(0)= x_0\), \(x(\theta)= \varphi(\theta)\). The quadratic cost functional to be minimized is \[ I(u)= x'(t_1) L(x(t_1))+ \int^{t_1}_0 x'M(t) x(t) dt+ \int^{t_1}_0 N(t) u^2(t) dt. \] Here, the prime denotes transposition. The trajectories obey the condition: \(x(\tau= t_1)= g(\tau= t_1)\). \(N\), \(M\) are piecewise continuous matrices, \(M\geq 0\), \(N>0\). It was shown by \textit{R. Bellman} and \textit{K. L. Cooke} [``Differential-difference equations'' (1963; Zbl 0105.06402)] that the solution may be written in the form \[ x(t)= K(t) x_0+ \int^0_h K(t-\tau-h) A_1(\varphi(\tau)) d\tau. \] This result is the starting point of the author's arguments. He offers a controllability criterion first for the real number field, then a stronger result over the complex number field, using his earlier results, and also some arguments, and statements indicating the importance of this problem, which can be traced to the \textit{M. N. Krasovskij's} 1968 classical work [``Theory of control motion: Linear systems'' (1968; MR 41\(\#\)96.22)]. Finally, he offers an algorithm leading to an exact solution of the minimization problem. The paper is a good example how one can elegantly use classical analysis to prove rather difficult problems of control theory.