UDC 519.25:62-501

MATHEMATICS

V. F. DEM’YANOV

ON A NECESSARY CONDITION FOR AN EXTREMUM IN CONTROL PROBLEMS WITH AFTEREFFECT

(Presented by Academician L. S. Pontryagin on 14 V 1965)

Let (U) be a convex, bounded, weakly closed class of summable real (r)-dimensional vector-functions on ([0,T]) ((T>0) fixed). The class (U) is called the class of controls, and a vector-function
[
u(t)=\bigl(u^1(t),\ldots,u^r(t)\bigr)\in U
]
is called a controlling vector-function or a control. For example, as (U) one may consider classes of summable (r)-dimensional vector-functions on ([0,T]) satisfying on ([0,T]) one of the following constraints:
[
|u^i(t)|\leq \alpha_i(t);\qquad \alpha_i(t)\geq 0;\qquad i=1,\ldots,r;\qquad t\in[0,T];
\tag{1}
]
[
u^(t)N(t)u(t)\leq \beta(t);\qquad \beta(t)\geq 0;\qquad t\in[0,T];
\tag{2}
]
[
\int_0^T u^{i2}(t)\,dt\leq C_i;\qquad 0<C_i<\infty;\qquad i=1,\ldots,r;
\tag{3}
]
[
\int_0^T u^(t)N(t)u(t)\,dt\leq C;\qquad 0<C<\infty,
\tag{4}
]
where (\alpha_i(t)) ((i=1,\ldots,r)), (\beta(t)) are piecewise continuous functions bounded on ([0,T]); (N(t)) is an (r\times r) matrix, positive definite on ([0,T]), with piecewise continuous entries bounded on ([0,T]); the asterisk denotes transposition.

We consider the problem of finding the minimum on (U) of a certain functional (I(u)), described below. A general problem of this kind was considered, in particular, in (1).

On ([0,T]) a system of (n) differential equations with variable delay is given:
[
dx(t)/dt=\dot{x}(t)=f\bigl(x(t),x(t-h_1(t)),u(t),t\bigr);
\tag{5}
]
[
x(0)=x_0(t)\qquad \text{for } t\in[-h_1(0),0].
\tag{6}
]
The function (v(t)=t-h_1(t)) is a strictly increasing continuously differentiable real function on ([0,T]),
[
0<h_1(t)<\infty\qquad \text{for } t\in[0,T];\qquad \min_{t\in[0,T]} h_1(t)>0.
]
Then there exists an inverse function (t=r_1(v)), which is also a strictly increasing continuously differentiable real function on ([-h_1(0),\,T-h_1(T)]); (u\in U), (f(x,y,u,t)=(f^1,\ldots,f^n)) is an (n)-dimensional real vector-function, continuous in (x^i,y^j,u^k,t) and continuously differentiable in (x^i,y^j,u^k) ((i,j=1,\ldots,n;\ k=1,\ldots,r)) in the domain of admissible values (x^i,y^j,u^k), determined by the class of controls (U), the system (5), (6), and the initial vector-function (x_0(t)), given and continuous on ([-h_1(0),0]).

By (x(t,u)=(x^1,\ldots,x^n)) we denote the solution of system (5) with initial conditions (6).

A functional is given

[
I(u)=\int_0^T g\bigl(x(t,u),x(t-h_2(t),u),u(t),t\bigr)\,dt,
\tag{7}
]

where (\nu(t)\equiv t-h_2(t)) is a strictly increasing continuously differentiable real function on ([0,T]); (0\leq h_2(t)<\infty) for (t\in[0,T]). Let (t=r_2(\nu)) be the function inverse to (\nu(t)) (it also is strictly increasing and continuously differentiable on ([-h_2(0),\,T-h_2(T)])); (g(x,y,u,t)) is a scalar real function continuous in (x^i,y^j,u^k,t) and continuously differentiable in (x^i,y^j,u^k) ((i,j=1,\ldots,n;\ k=1,\ldots,r)) in the domain of admissible values of (x^i,y^j,u^k) and for (t\in[0,T]).

We assume that if (h_2(0)>h_1(0)), then (x(t)) is prescribed and continuous on ([-h_2(0),-h_1(0)]).

From the class of controls (U) it is required to find a control (u\in U) such that

[
I(u)=\min_{v\in U} I(v).
\tag{8}
]

Such a (u\in U) is called an optimal control.

Theorem. In order that the control (u\in U) give the functional (7) its minimal value on (U), it is necessary, and in the case of convexity of (I(u)) also sufficient, that

[
\min_{v\in U}\int_0^T \sum_{i=1}^r
\left[\left(\frac{\partial f(\tau)}{\partial u^i}\right)^*\psi(\tau)
+\frac{\partial g(\tau)}{\partial u^i}\right]
\bigl(v^i(\tau)-u^i(\tau)\bigr)\,d\tau=0,
\tag{9}
]

where (\psi(\tau)) is an (n)-dimensional vector-function satisfying the system of differential equations

[
\dot\psi(\tau)=
\begin{cases}
-\left(\dfrac{\partial f(\tau)}{\partial x}\right)^\psi(\tau)
-\left(\dfrac{\partial f(r_1(\tau))}{\partial y}\right)^
\psi(r_1(\tau))\,\dot r_1(\tau)-C(\tau),
& \text{for } \tau\in[0,T-h_1(T)],\[1.2em]
-\left(\dfrac{\partial f(\tau)}{\partial x}\right)^*\psi(\tau)-C(\tau),
& \text{for } \tau\in[T-h_1(T),T],
\end{cases}
\tag{10}
]

[
\psi(T)=0;
\tag{11}
]

[
f(\tau)=f\bigl(x(\tau,u),x(\tau-h_1(\tau),u),u(\tau),\tau\bigr);
]

[
g(\tau)=g\bigl(x(\tau,u),x(\tau-h_2(\tau),u),u(\tau),\tau\bigr);
]

[
\partial g/\partial x=(\partial g/\partial x^1,\ldots,\partial g/\partial x^n);
\qquad
\partial g/\partial y=(\partial g/\partial y^1,\ldots,\partial g/\partial y^n);
]

[
\partial f/\partial x=
\begin{pmatrix}
\partial f^1/\partial x^1 & \cdots & \partial f^1/\partial x^n\
\cdots & \cdots & \cdots\
\partial f^n/\partial x^1 & \cdots & \partial f^n/\partial x^n
\end{pmatrix};
\qquad
\partial f/\partial y=
\begin{pmatrix}
\partial f^1/\partial y^1 & \cdots & \partial f^1/\partial y^n\
\cdots & \cdots & \cdots\
\partial f^n/\partial y^1 & \cdots & \partial f^n/\partial y^n
\end{pmatrix};
]

[
C(\tau)=
\begin{cases}
\dfrac{\partial g(\tau)}{\partial x}
+\dfrac{\partial g(r_2(\tau))}{\partial y}\,\dot r_2(\tau),
& \text{for } \tau\in[0,T-h_2(T)],\[1.2em]
\dfrac{\partial g(\tau)}{\partial x},
& \text{for } \tau\in[T-h_2(T),T].
\end{cases}
]

The proof of this theorem is carried out using the results obtained in (1).

If delay is absent both in system (5) and in functional (7) (i.e. (h_1(t)\equiv h_2(t)\equiv 0)), then the necessary condition (9), as is easy to see, is preserved, and the system for (\psi(\tau)) instead of (10) will have the form

[
\dot\psi(\tau)=-(\partial f(\tau)/\partial x)^*\psi(\tau)-\partial g(\tau)/\partial x,
\tag{12}
]

[
\psi(T)=0.
\tag{13}
]

It is clear that in this case, for classes of controls with constraints of the form (1) and (2), condition (9) is a “linearization” of L. S. Pontryagin’s “maximum principle” ($^2$). Verifying the fulfillment of condition (9) is simpler than verifying the fulfillment of the “maximum principle.” Moreover, the necessary condition (9) is valid for classes of controls $U$ of a more general form than the “maximum principle” (in particular, for classes of controls with constraints of integral type (3), (4)).

Similarly, for the case of a constant delay

[
h_1(t) \equiv h_2(t) \equiv h = \mathrm{const}
]

a necessary condition has been obtained which is (for classes of controls of the form (1), (2)) a linearization of the “maximum principle” (($^2$), pp. 236–250). For the case of a variable delay (and classes of controls of the form (1), (2)), a necessary condition analogous to the “maximum principle” was obtained by Yu. F. Kazarinov ($^3$).

A necessary condition analogous to (9) can be obtained for systems of integro-differential equations.

To find a control satisfying condition (9), one may apply one of the methods of successive approximations proposed in ($^1$).

Leningrad State University
named after A. A. Zhdanov

Received
25 II 1965

REFERENCES

$^1$ V. F. Demyanov, A. M. Rubinov, Vestn. LGU, Ser. Math., Mech., Astr., No. 19 (1964).
$^2$ L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Moscow, 1961.
$^3$ Yu. F. Kazarinov. Collected volume: Computational Methods, issue 3, Leningrad, 1965.