The deviation of a functional from the optimal value under chattering exponentially decays as the number of switchings tends to infinity (Q2703870)

The authors consider the problem of minimization of a quadratic criterion NEWLINE\[NEWLINE J=\frac{1}{2}\int_0^\infty x^2(t) dt NEWLINE\]NEWLINE on the trajectories of a system that can be interpreted as a nonlinear perturbation of a linear system NEWLINE\[NEWLINE \frac{d^n}{dt^n}x(t)=h_0+ u(t)h_1 \quad, |u(t)|\leq 1. NEWLINE\]NEWLINE It is known, that at transition from regular to singular mode of control, the number of switching points of the optimal control is infinite on a finite time interval, is called chattering. NEWLINENEWLINENEWLINEA practically important and elegant result is obtained: the authors propose a procedure of replacing the optimal chattering control with a suboptimal one with a finite number of switching points and provide an estimate of the respective deviation of the value of the cost functional from the optimal value. The control is constructed in the following way. First, an optimal arc with a fixed number of switching points is constructed. Then it is appended by a trajectory reaching the origin in a finite time. This part of the trajectory may be not optimal, but the time of reaching the origin satisfies an estimate reported in [\textit{L. A. Manila}, Fundam. Prikl. Mat. 2, No. 2, 411-417 (1996; Zbl 0901.49001)]. The resulting suboptimal control has a finite number of switching points and the deviation of the functional from the optimal value decays exponentially as the number of switching points increases.