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.