Regularization and Stabilization of Randomly Switching Dynamic Systems

This work concerns regularization (or explosion suppression) and stabilization of randomly switching dynamic systems by selecting suitable input functions. Starting with a randomly switching system modulated by a continuous-time Markov chain, our aim is to find a feedback control (input function) so that the system becomes stable. But before addressing the stabilization issue, another problem needs to be resolved first. Because of the fast growth in the continuous state variable, the solution of the system has a finite explosion time with probability one. To ensure the existence of global solutions, a feedback control (a noise perturbation) is added to make the system regular. Then another feedback control (a second perturbing noise) is added to ensure the stability of the resulting systems. Owing to the nonlinearity, closed-form solutions are often virtually impossible to obtain. Thus, a discrete-time approximation algorithm is constructed. Under simple conditions, a suitably interpolated sequence of the discrete-time algorithm is shown to converge to the limit switching diffusion. Finally, regularization and stabilization of the approximating sequence are dealt with.