Stabilization of linear dynamic systems by low-inertia controls. I: Associated problem of optimal control (Q1290748)

The authors consider the problem of damping linear dynamic systems on a finite interval \(T=[0,\vartheta]\) by low-inertia controls of minimal intensity. The behavior of the stationary system is described by the equation \[ \dot x=Ax+bu,\quad x(0)=x^\star, \] where \(x\) is an \(n\)-vector of phase state, \(u\) is a scalar control, \(x^\star\neq 0.\) The control actions \(u(t)\), \(t\in T,\) are chosen in the class of continuous functions having a piecewise-continuous derivative satisfying \(|\dot u(t)|\leqslant 1/\mu,\) here \(\mu\) is a small positive parameter. Such functions are referred to as accessible controls. The accessible controls are called admissible ones if the terminal constraint \(x(\vartheta) = 0\) holds. The quality of the control process is evaluated by the functional \(J(u)=\max_{t\in T}| u(t)|.\) The problem of minimization of this functional in the class of admissible controls is considered. The control, on which the minimum is reached, is called an optimum one. The accessible control \(u(t,\mu)\) is called asymptotically \(k\)-admissible, if for the trajectory \(x(t,\mu)\) it generates, the condition \(x(\vartheta,\mu)=0(\mu^{k+1})\) is fulfilled. The admissible or asymptotically \(k\)-admissible control is called asymptotically \(s\)-optimum, if the deviation of the criterion of quality \(J(u)\) from the optimum value is of order \(o(\mu^{s+1}).\) An algorithm enabling for a given natural number \(s\) to construct the asymptotically \(s\)-admissible \(s\)-optimum control is offered.