Stabilizing Regulator in One Class of Continuous System with Control Constraints

The article considers the problem of constructing a stabilizing control for one class of nonlinear continuous systems with constraints on controls, in which the right-hand sides are linear in state and formally linear in control, but the coefficients of the matrix under control can be dependent on the coordinates of the state vector. Feedback in the work is built by introducing a quadratic quality criterion with a weight matrix for a state with coefficients depending on the coordinates of the state vector, i.e. by solving an auxiliary optimal control problem, where the SDRE approach can be applied in which the corresponding matrix Riccati equation is solved, the matrices in which can be state-dependent. But for this class of control problems, the article proposes an approach to constructing a stabilizing control by the extension method of V.F. Krotov, which makes it possible not to recalculate the matrix of gains in the feedback loop at each step of calculating the trajectories of the closed-loop system. In this work, it is possible to prove the asymptotic stability of the zero equilibrium position of a closed-loop system by constructing the Lyapunov function, which is the sum of the classical positive definite quadratic form for a linear control system. At the end of the work, a numerical experiment is presented that illustrates the statements using the example of stabilization in a three-sector economic system.