Estimates of the number of switchings of a relay optimal control for a certain class of bilinear controlled systems (Q1335812)

The problem of finding the maximal value of \(\langle p, x(1)\rangle\) subject to \[ \dot x= Axu+ Cu+ D, \quad 0\leq t\leq 1, \qquad x(0)= x_ 0, \quad a\leq u(t)\leq b, \] is considered. It is assumed that \(a>0\) and the admissible controls are scalar functions. It is obtained that the optimal relay control is a piecewise constant function having at most \(2M\) switching points, where \(M\) is the number of local maxima of \[ \varphi(z)= \langle p, \exp(Az)D \rangle, \qquad z\in [0,b]. \] Since the function \(\varphi(z)\) is analytic, the number \(M\) is finite. An algorithm to calculate the optimal control via finite dimensional optimization procedure is proposed. Using this algorithm, an explicit formula of the optimal control for a system governed by Kirchhoff's law equations is derived. A similar estimate of the number of switchings is derived for the problem of maximization of a linear terminal functional on the set of solutions of a boundary problem for the diffusion equation with the diffusion coefficients as a control. It is asserted that for this problem the above mentioned algorithm is applicable, too.