MATHEMATICS

V. I. ZUBOV

ON A PROBLEM OF OPTIMAL STABILIZATION

(Presented by Academician N. N. Krasovskii, 16 VI 1969)

In the present note a problem of optimal stabilization is considered in the form of a differential game, and conditions for existence and a method for constructing optimal controls in analytic form are given.

Let us first consider the linear system of differential equations

[
\dot X = PX + QU
\tag{1}
]

and the functional

[
I(X_0,U)=\int_0^\infty W^{(2)}\,dt,
\tag{2}
]

where

[
W^{(2)} = X^AX + X^BU + U^B^X + U^*CU.
]

We shall assume that the elements of the matrices (A, B, C, C^{-1}, P), and (Q) are real, continuous, and bounded functions of time, given for (t>0).

Put (X=(X_1,X_2)) and (U=(U_1,U_2)). We shall assume that the vectors (X_i, U_i) have dimensions (n_i) and (r_i), respectively. In accordance with this the matrix (C) may be represented in the form

[
C=\begin{pmatrix}
C_1 & D\
D^* & C_2
\end{pmatrix}.
]

We shall make an assumption that is essential for what follows. Namely, we shall suppose that the quadratic form (U_1^C_1U_1) is positive definite, while (U_2^C_2U_2) is negative definite.

Definition 1. A control (U(t,X)) is called admissible if: 1) (U(t,X)=M(t)\cdot X), where (M(t)) is a matrix with real, continuous, and bounded coefficients, given for (t>0); 2) system (1) under the control (U=M(t)\cdot X) has a uniformly asymptotically stable equilibrium position (X=0) of exponential type.

Definition 2. An admissible control (U_0=M_0(t)\cdot X) is called optimal if

[
I(X_0,U_0)=\min_{U_1}\max_{U_2} I(X_0,U)
=\max_{U_2}\min_{U_1} I(X_0,U)
]

for every initial vector (X_0). Here (\min\max) and (\max\min) are taken over all such controls (U_1) and (U_2) that form an admissible control (U=(U_1,U_2)).

Remark. The optimal control (U_0) may thus be regarded as an equilibrium situation in a differential game of two persons (X_1, X_2), whose sets of admissible strategies are given by Definition 1.

Theorem 1. In order that an optimal control (U_0) exist, it is necessary and sufficient that there exist a real continuous-

a bounded matrix (\theta), defined for (t>0), satisfying the equation

[
\dot{\theta}+\theta Q C^{-1}Q^{}\theta+\theta(P-QC^{-1}B^{})+
(P-QC^{-1}B^{})^{}\theta-A+BC^{-1}B^{*}=0
\tag{3}
]

and such that the control (C^{-1}(Q^{}\theta-B^{})\cdot X) is admissible. In this case (U_{0}=C^{-1}(Q^{}\theta-B^{})\cdot X).

Let us next consider the nonlinear system of differential equations

[
\dot{X}=PX+QU+\sum_{m=2}^{\infty}F^{(m)}=G(t,X,U),
\tag{4}
]

and the functional

[
\int_{0}^{+\infty}\sum_{m=2}^{\infty}W^{(m)}=I(X_{0},U).
\tag{5}
]

Here (F^{(m)}) and (W^{(m)}) are homogeneous forms of degree (m) with respect to the components of the vectors (X) and (U), with real, continuous, bounded coefficients, defined for (t>0). We shall henceforth assume that the series (W=\sum W^{(m)}), (F=\sum F^{(m)}) converge in some fixed neighborhood of the point (X=0,\ U=0) uniformly with respect to (t>0).

Definition 3. A control (U(t,X)) is called admissible if:

[
1)\quad U(t,X)=\sum_{m=1}^{\infty}U^{(m)};
]

2) (U^{(1)}(t,X)) is an admissible control in the sense of Definition 1.

Here we assume that the series representing the admissible control converges uniformly with respect to (t\ge 0) in some fixed neighborhood of the point (X=0).

Definition 4. An admissible control (U_{0}(t,X)) is called optimal for system (4) with respect to the functional (5) if there exists some neighborhood of the point (X=0) such that for any initial value (X_{0}) from this neighborhood one has

[
I(X_{0},U_{0})=\min_{U_{1}}\max_{U_{2}} I(X_{0},U)
=\max_{U_{2}}\min_{U_{1}} I(X_{0},U).
]

Here, as above, (\min\max) and (\max\min) are computed for those (U_{1}) and (U_{2}) that provide an admissible control (U=(U_{1},U_{2})) in the sense of Definition 3.

Theorem 2. If there exists a real continuous bounded solution (\theta) of equation (3), defined for (t>0), such that the control (M_{0}\cdot X) is admissible in the sense of Definition 1,

[
M_{0}=C^{-1}(Q^{}\theta-B^{}),
]

and the system (\dot{X}=P_{0}X,\ P_{0}=P+QM_{0}), is proper, then there exists an optimal control

[
U_{0}(t,X)=\sum_{m=1}^{\infty}U_{0}^{(m)}
]

for system (4) with respect to the functional (5), representable in the form of series uniformly convergent for (t\ge 0). Here, as above, (U_{0}^{(m)}) are homogeneous forms of degree (m) in the components of the vector (X).

Consider the system of equations

[
\frac{\partial \lambda}{\partial t}
+
\frac{\partial \lambda}{\partial x}\, G(t, X, U)
=
-
\frac{\partial G^*}{\partial x}\,\lambda
+
\frac{\partial W}{\partial x},
\tag{6}
]

[
\frac{\partial W}{\partial U}
=
\lambda^* \frac{\partial G}{\partial U}.
\tag{7}
]

Here (\lambda) is a vector of dimension (n); the partial derivatives with respect to the vector (X) and the vector (U) are defined in the natural way as matrices of the corresponding dimensions.

Theorem 3. If the conditions of Theorem 2 are satisfied, then the system (6), (7) has a unique solution in the form of convergent series

[
\lambda = \sum \lambda^{(m)}, \qquad U = \sum U^{(m)},
\tag{8}
]

[
\lambda^{(1)} = -2\theta X,\quad U^{(1)} = M_0 \cdot X,
]

uniformly with respect to (t \ge 0), and the series (8) gives the desired optimal control.

Received
24 IV 1969