Optimal guaranteed control of linear systems under disturbances.

*(English. Russian original)*Zbl 0956.49017
Math. Notes 60, No. 2, 147-152 (1996); translation from Mat. Zametki 60, No. 2, 198-205 (1996).

The author considers the following linear systems
\[
\dot x(t)=A(t)x(t)-B(t)u(t)-D(t)w(t)+C(t)v(t), \tag{1}
\]
\(t\in [t^0,\theta], \;x(t^0)=x^0\), where \(A(t), B(t), C(t), D(t)\) are continuous matrices of dimensions \(n\times n\), \(n\times m\), \(n\times l\), \(n\times k\), respectively.

Parameters \(u\) and \(w\) are controls, and \(v\) is a disturbance. It is assumed that \[ u(t)\in U(t), \tag{2} \] where \(t\mapsto U(t)\subset R_m\) is a continuous convex- and compact-valued mapping.

The performance (cost) functional has the form \[ I=\gamma(x(\theta))+\int_\theta^{t^0} ( w_T (t)G(t)w(t)v_T (t)H(t)v(t)) dt, \tag{3} \] where \[ \;\gamma(x)=\min_{y\in M^0} (x-y)_T R^0 (x-y), \tag{4} \] \(G(t), H(t)\) are symmetric positive semidefinite continuous matrices, \(R^0\) is a positive definite matrix, \(M^0 \subset R_n\) is the convex closed target set.

The aim of controls is minimizing the cost functional, the disturbance tries to maximize the cost functional.

It is proven that the differential game has the value function, i.e., there exists the equilibrium in the classes of feedbacks: Borel measurable controls \(u(t,x)\) plus continuous controls \(w(t,x)\) and continuous disturbances \(v(t,x)\).

In the paper, simple formulas for calculating the value function and optimal guaranteed feedbacks are obtained.

Parameters \(u\) and \(w\) are controls, and \(v\) is a disturbance. It is assumed that \[ u(t)\in U(t), \tag{2} \] where \(t\mapsto U(t)\subset R_m\) is a continuous convex- and compact-valued mapping.

The performance (cost) functional has the form \[ I=\gamma(x(\theta))+\int_\theta^{t^0} ( w_T (t)G(t)w(t)v_T (t)H(t)v(t)) dt, \tag{3} \] where \[ \;\gamma(x)=\min_{y\in M^0} (x-y)_T R^0 (x-y), \tag{4} \] \(G(t), H(t)\) are symmetric positive semidefinite continuous matrices, \(R^0\) is a positive definite matrix, \(M^0 \subset R_n\) is the convex closed target set.

The aim of controls is minimizing the cost functional, the disturbance tries to maximize the cost functional.

It is proven that the differential game has the value function, i.e., there exists the equilibrium in the classes of feedbacks: Borel measurable controls \(u(t,x)\) plus continuous controls \(w(t,x)\) and continuous disturbances \(v(t,x)\).

In the paper, simple formulas for calculating the value function and optimal guaranteed feedbacks are obtained.

Reviewer: N.Subbotina (Sverdlovsk)

##### MSC:

49K35 | Optimality conditions for minimax problems |

49N10 | Linear-quadratic optimal control problems |

49N70 | Differential games and control |

93C73 | Perturbations in control/observation systems |

91A23 | Differential games (aspects of game theory) |

##### Keywords:

optimal control under disturbances; minimax control; optimal guaranteed control; geometric control constraints; performance functional
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\textit{G. E. Ivanov}, Math. Notes 60, No. 2, 147--152 (1996; Zbl 0956.49017); translation from Mat. Zametki 60, No. 2, 198--205 (1996)

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##### References:

[1] | L. S. Pontryagin, ”Linear differential games of pursuit,”Mat. Sb. [Math. USSR-Sb.],112, No. 3, 307–330 (1980). · Zbl 0445.90118 |

[2] | N. N. Krasovskii,Control of a Dynamical System [in Russian], Nauka, Moscow (1985). |

[3] | V. M. Kein, V. S. Patsko and V. L. Turova, ”The aircraft landing problem under changes of wind direction,” in:Control in Dynamical Systems [in Russian], Sverdlovsk, (1990), pp. 52–64. |

[4] | A. M. Taras’ev, A. A. Uspenskii and V. N. Ushakov, ”Appproximate construction of the position absorption set in the linear problem of approach to a convex target in \(\mathbb{R}\)3,” in:Control in Dynamical Systems [in Russian], Sverdlovsk (1990), pp. 93–100. |

[5] | A. N. Kolmogorov and S. V. Fomin,Elements of the Theory of Functions and of Functional Analysis [in Russian], Nauka, Moscow (1989). · Zbl 0672.46001 |

[6] | B. I. Arkin and V. L. Levin, ”The convexity of the values of vector integrals, the theorems of measurable choice, and variational problems,”Uspekhi Mat. Nauk [Russian Math. Surveys],27, No. 3, 26 (1972). · Zbl 0256.49025 |

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