Algorithm for finding feedback in a problem with constraints for one class of nonlinear control systems

For a continuous nonlinear control system on a finite time interval with control constraints, where the right-hand side of the dynamics equations is linear in control and linearizable in the vicinity of the zero equilibrium position, we consider the construction of a feedback according to the Kalman algorithm. For this, the solution of an auxiliary optimal control problem with a quadratic functional is used by analogy with the SDRE approach.
Since this approach is used in the literature to find suboptimal synthesis in optimal control problems with a quadratic functional with formally linear systems, where all coefficient matrices in differential equations and criteria can contain state variables, then on a finite time interval it becomes necessary to solve a complicated matrix differential Riccati equations, with state-dependent coefficient matrices. This circumstance, due to the nonlinearity of the system, in comparison with the Kalman algorithm for linear-quadratic problems, significantly increases the number of calculations for obtaining the coefficients of the gain matrix in the feedback and for obtaining synthesis with a given accuracy. The proposed synthesis construction algorithm is constructed using the extension principle proposed by V. F. Krotov and developed by V. I. Gurman and allows not only to expand the scope of the SDRE approach to nonlinear control problems with control constraints in the form of closed inequalities, but also to propose a more efficient computational algorithm for finding the matrix of feedback gains in control problems on a finite interval. The article establishes the correctness of the application of the extension principle by introducing analogs of the Lagrange multipliers, depending on the state and time, and also derives a formula for the suboptimal value of the quality criterion. The presented theoretical results are illustrated by calculating suboptimal feedbacks in the problems of managing three-sector economic systems.