LP-stabilization problem for linear stochastic control systems with multiplicative noise

Abstract

For the deterministic case, a linear controlled system is alwayspth order stable as long as we use the control obtained as the solution of the so-called LQ-problem. For the stochastic case, however, a linear controlled system with multiplicative noise is not alwayspth mean stable for largep, even if we use the LQ-optimal control. Hence, it is meaningful to solve the LP-optimal control problem (i.e., linear system,pth order cost functional) for eachp. In this paper, we define the LP-optimal control problem and completely solve it for the scalar case. For the multidimensional case, we get some results, but the general solution of this problem seems to be impossible. So, we consider thepth mean stabilization problem more intensively and give a sufficient condition for the existence of apth mean stabilizing control by using the contraction mapping method in a Hilbert space. Some examples are also given.