Optimum stabilization of control systems under constraints on the output variable (Q1342550)

The authors consider the following linear-quadratic optimal control problem: \[ {{dx} \over {dt}}= Ax+ bu, \quad x(0)= a\quad (0\leq t<\infty), \qquad |x(\cdot)|\in L_2 (0, \infty), \quad |u(\cdot)|\in L_2 (0, \infty), \] \[ \alpha_j\leq y(t_j)\leq \beta_j, \quad j=1, \dots, k, \qquad \text{where } y(t)= c^* x(t), \tag{1} \] \[ I=\int^\infty_0 Q(x(t) u(t))dt\to \min. \] Here \(A\), \(b\), \(c\) are constant real matrices (resp. vectors) with \(c\neq 0\), and \(Q\) is a quadratic form. Using known results on the corresponding problem without the constraints (1), the authors establish a sufficient condition for the existence of optimal control to the above problem. An algorithm for calculating optimal controls is also discussed.