Optimization of control in a nonlinear stochastic system by a local criterion (Q1390542)

The author considers the problem of optimal control for nonlinear stochastic systems. The state and output equations have the form \[ dx(t)=\bigl[ A(x,t)+B(x,t)u+ w(t)\bigr]dt+ N(x,t)d\xi(t)\tag{1} \] \[ dy=C(t)xdt+ d\eta(t) \tag{2} \] where \(x\in R^n\), \(y\in R^p\) and \(u\in R^r\) are state, observation and control vectors, respectively; \(A(x,t)\), \(w(t)\) are vector deterministic functions, \(B(x,t)\), \(N(x,t)\) and \(C(t)\) are matrix deterministic functions; \(\xi(t)\) and \(\eta(t)\) are Gaussian-centered white noises with zero mean values and intensity matrices \(G(t)\) and \(Q(t)\), respectively. The initial vector of the state \(x_0\) is a random variable with given mean value and variance matrix. The vector of control is bounded; i.e., \(u\in U\), \(| u_i|\leq u_{i0}\), \(i=1, \dots,r\). The criterion \(I_{0t}\) is taken in the form of the generalized square function. The problem consists in the choice of the optimal control vector \(u\) for the stochastic system (1) from the condition \[ u={\underset {u\in U} {\text{argmin}}} E\bigl[I_{0t}\mid y(\tau),t_0\leq \tau\leq t\bigr] \tag{3} \] at an arbitrary moment in the observation of the vector \(Y(\tau)\). The solution was found by a combination of Taylor or statistical linearization and Kalman filtering. An example illustrates the results obtained.