### A NUMERICAL METHOD FOR SOLVING LINEAR–QUADRATIC CONTROL PROBLEMS WITH CONSTRAINTS

#### Abstract

The paper is devoted to the optimal control problem for a linear system with integrally constrained control function. We study the problem of minimization of a linear terminal cost with terminal constraints given by a set of linear inequalities. For the solution of this problem we propose two-stage numerical algorithm, which is based on construction of the reachable set of the system. At the first stage we find a solution to finite–dimensional optimization problem with a linear objective function and linear and quadratic constraints. At the second stage we solve a standard linear–quadratic control problem, which admits a simple and effective solution.

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- Anan’ev B.I. Motion correction of a statistically uncertain system under communication constraints // Automation and Remote Control. 2010. Vol. 71, no. 3. P. 367–378. DOI: 10.1134/S0081543810060039
- Antipin A.S., Khoroshilova E.V. Linear programming and dynamics // Trudy Inst. Mat. i Mekh. UrO RAN. 2013. Vol. 19, no. 2. P. 7–25. [in Russian]
- Antipin A.S., Khoroshilova E.V. Linear programming and dynamics // Ural Mathematical Journal, 2015. Vol. 1, no. 1. P. 3–19. DOI: 10.15826/umj.2015.1.001
- Arutyunov A.V., Magaril-Il’yaev G.G. and Tikhomirov V.M. Pontryagin maximum principle. Proof and applications. Moscow: Factorial press, 2006. 124 p. [in Russian]
- Dar’in A.N., Kurzhanski A.B. Control under indeterminacy and double constraints // Differential Equations. 2003. Vol. 39, no. 11. P. 1554–1567.
- Goberna M.A.,Lopez M.A.Linear semi-infinite programming theory: An updated survey // European J. of Operational Research. 2002. Vol. 143, Issue 2. P. 390–405.
- Gusev M.I. On optimal control problem for the bundle of trajectories of uncertain system // Lecture Notes in Computer Sciences. Springer. LNCS 5910. 2010. P. 286–293. DOI: 10.1007/978-3-642-12535-5_33
- Krasovskii N.N. Theory of Control of Motion. Moscow: Nauka, 1968. 476 p. [in Russian]
- Kurzhanski A.B. Control and Observation under Conditions of Uncertainty. Moscow: Nauka, 1977. 392 p. [in Russian]
- Lee E.B., Marcus L. Foundations of Optimal Control Theory. Jhon Willey and Sons, Inc., 1967. 124 p.
- Martein L., Schaible S. On solving a linear program with one quadratic constraint // Rivista di matematica per le scienze economiche e sociali. 1988. P. 75–90.
- Ushakov V.N. Extremal strategies in differential games with integral constraints // J. of Applied Mathematics and Mechanics. 1972. Vol. 36, no. 1. P. 15–23.
- Ukhobotov V.I. On a class of differential games with an integral constraint // J. of Applied Mathematics and Mechanics. 1977. Vol. 41, no. 5. P. 838–844.

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