Stability of discrete systems with random structure under steadily acting disturbances (Q788688)

The author considers a nonlinear control system described by a stochastic difference equation \(x_{n+1}=f(x_ n,y_ n)+G(x_ n,y_ n)v_ n (n=0,1,...)\), where \(x_ n\) is an m-dimensional state vector, f(x,y) is an m-dimensional vector function, G(x,y) is an (\(m\times k)\)-dimensional matrix whose elements are functions of x and y, \(v_ n\) is k-dimensional Gaussian discrete white noise with a unit covariance matrix and \(y_ n\) is a scalar Markov chain with a known transition matrix \([P_{ij}] (i,j=1,2,...,0)\), \(P_{ij}=P\{y_{n+1}=j| y_ n=i\}.\) Under suitable assumptions the author gives conditions for p-stability of the system (1). He also considers the problem of optimal stabilization of a linear system with additive and multiplicative noise, further the problem of nonlinear stochastic Lur'e type systems. In both cases using the Lyapunov approach he gives sufficient conditions for stabilizability. An example is also given.