System shape optimization and stabilization of controlled quasi-linear stochastic systems that operate on an infinite time interval

Abstract

The necessary conditions in the stabilization and optimization problem for a stationary quasi-linear stochastic system in continuous time, with its matrices depending on a vector parameter to be chosen, i.e., the optimization problem for the system shape, are obtained. An equivalent deterministic problem is stated and a numerical method to solve it using the analytical formula obtained for the criterion gradient, which is the function of a finite number of variables, is proposed. The optimization problem for an output-controlled system is a particular case, sufficient optimality conditions are obtained for it in the case that the complete information of the state is available. Optimality conditions are found for the proportional---integral---derivative controller in the quasi-linear stochastic system. These optimality conditions are applied to the optimal control problem for a small unmanned aerial vehicle moving in a disturbed atmosphere.