Mathematics

R. I. Zarosskii

On the Question of Necessary and Sufficient Conditions for an Absolute Minimum

(Presented by Academician L. S. Pontryagin on 2.III.1965)

We consider the problem of the absolute minimum of the functional

\[ I(x,u)=\int_A f^0(t,x,u)\,dt+F(x(\tau)), \tag{1} \]

where \(t=(t^1,\ldots,t^m)\), \(x=(x^1,\ldots,x^n)\), \(u=(u^1,\ldots,u^r)\) are elements of the vector spaces \(T\), \(X\), and \(U\), respectively, and \(A\) is a closed domain in the space \(T\), bounded by a continuous, piecewise-smooth hypersurface \(S\), with \(t=\tau\) on \(S\). Define the set \(E\) of all pairs of functions \(x(t)\), \(u(t)\), defined on the domain \(A\), such that \(x^i(t)\) \((i=1,\ldots,n)\) are continuous and have bounded partial derivatives for \(t\in A\), while \(u^k(t)\) \((k=1,\ldots,r)\) are continuous everywhere on \(A\), except for a finite number of points, and at the points of discontinuity have bounded partial derivatives. Next, define the set \(D\subset E\) of pairs of functions \(x(t)\), \(u(t)\) satisfying, in addition to the indicated conditions, the system of \((nm)\) differential equations in partial derivatives

\[ \frac{\partial x^i}{\partial t^j}=f_j^i(t,x,u) \quad (i=1,\ldots,n;\ j=1,\ldots,m). \tag{2} \]

The functions \(f_j^i(t,x,u)\) and the function \(f^0(t,x,u)\) are continuous together with all first-order partial derivatives for \(t\in A\), \(x,u\in E\). The functional \(F(x(\tau))\) is defined on the set of values \(x(\tau)\) of the function \(x(t)\) on the surface \(S\) and is bounded. The set of admissible values \(x(t)\) and the set of admissible controls \(u(t)\), considered for fixed \(t\in A\), may be closed sets in the corresponding spaces \(X\) and \(U\).

The aggregate of points \(t\in A\) and the corresponding values \(x(t)\) at these points determines a certain set \(B\) in the \((n+m)\)-dimensional space \(T\times X\). Let \(B(t)\) be the “section” of the set \(B\) by some fixed value \(t\in A\). On the set \(B\) we shall consider continuous functions \(\varphi_j(t,x)\) \((j=1,\ldots,m)\) possessing continuous partial derivatives. Using Ostrogradsky’s formula and the relations (2), we have \((x,u\in D)\):

\[ \int_A \sum_{j=1}^m \left[ \sum_{i=1}^n \frac{\partial \varphi_j(t,x)}{\partial x^i}\, f_j^i(t,x,u) + \frac{\partial \varphi_j(t,x)}{\partial t^j} \right] dt = \int_S \sum_{j=1}^m \varphi_j(\tau,x)\cos(n,t^j)\,d\tau, \tag{3} \]

where \(n\) is the direction of the outward normal to the surface \(S\). If we now introduce into consideration the functions

\[ R(t,x,u)= \sum_{j=1}^m \left[ \sum_{i=1}^n \frac{\partial \varphi_j(t,x)}{\partial x^i}\, f_j^i(t,x,u) + \frac{\partial \varphi_j(t,x)}{\partial t^j} \right] - f^0(t,x,u), \tag{4} \]

\[ G(x(\tau))= F(x(\tau)) + \int_S \sum_{j=1}^m \varphi_j(\tau,x)\cos(n,t^j)\,d\tau, \tag{5} \]

then the functional (1) can, by virtue of (3), be represented in the form

\[ I(x,u)=G(x(\tau))-\int_A R(t,x,u)\,dt \qquad (x,u\in D). \tag{6} \]

The following generalized principle of optimality holds:

Theorem 1. Let there be a pair \(\bar x(t), \bar u(t)\in D\). In order that this pair minimize the functional (1) on the set \(D\), it is necessary and sufficient that there exist functions \(\varphi_j(t,x)\) \((j=1,\ldots,m)\) such that:

1) for all \(t\in A^*=A\setminus S\), except for a finite number of points,

\[ R(t,\bar x(t),\bar u(t))=\sup_{x,u\in E} R(t,x,u); \tag{7} \]

2) for all \(\tau\in S\)

\[ G(\bar x(\tau))=\inf_{x(\tau)\in B(\tau)} G(x(\tau)). \tag{8} \]

The sufficiency of the indicated conditions was considered by V. F. Krotov in \((^{1-3})\).

Necessity. Let \(P=P(t_0)\) be a fixed point of the domain \(A^*\), and let \(\delta\) be a nonnegative small parameter. Define the domain \(P_\delta\subset A^*\), bounded by a sphere with center at the point \(P\) and volume \(\delta\), and families of so-called needle variations (see, for example, \((^4)\)) of the functions \(x(t)\) and \(u(t)\):

\[ x_\delta(t)= \begin{cases} \bar x(t), & \text{for } t\in A^*\setminus P_\delta,\\ \widetilde x(t), & \text{for } t\in P_\delta, \end{cases} \qquad (\delta\ge 0), \tag{9} \]

\[ x_0(t)=\bar x(t); \]

\[ u_\delta(t)= \begin{cases} \bar u(t), & \text{for } t\in A^*\setminus P_\delta,\\ \widetilde u(t), & \text{for } t\in P_\delta, \end{cases} \qquad (\delta\ge 0), \tag{10} \]

\[ u_0(t)=\bar u(t), \]

where \(\widetilde x,\widetilde u\) is an arbitrary fixed pair of values from the set \(E\). It can be shown that the families (9), (10) are admissible in the sense of \((^4)\). Considering, on the totality of values \(t\in A,\ x,u\in E\), the \((n+m)\)-dimensional vector-function
\[ \Phi(t,x,u)\equiv f(t,x,u)-dx/dt, \]
we define on the set \(E\) the functional

\[ \Psi(x,u)=I(x,u)-L\Phi(t,x,u), \tag{11} \]

where \(L\) is some linear functional in the space of continuous vector-functions. Using the result obtained in \((^4)\), one may assert that for the functional (11) the condition

\[ \Psi'_\delta(x_\delta,u_\delta)\big|_{\delta=0}\ge 0 \tag{12} \]

is satisfied. The functional \(\Psi(x,u)\) can be represented in the form

\[ \Psi(x,u)=\int_A f^0(t,x,u)\,dt+F(x(\tau))-\int_A \left\{ \sum_{j=1}^{m}\sum_{i=1}^{n} \left[ f_j^i(t,x,u)-\frac{\partial x^i}{\partial t^j} \right] \frac{\partial \varphi_j(t,x)}{\partial x^i} \right\}\,dt, \]

where \(\varphi_j(t,x)\) \((j=1,\ldots,m)\) are functions continuous on \(B\) with continuous partial derivatives. Finding the increment

\[ \Delta\Psi(x_\delta,u_\delta)=\Psi(x_\delta,u_\delta)-\Psi(\bar x,\bar u) \]

and then computing \(\Psi'_\delta(x_\delta,u_\delta)\big|_{\delta=0}\), it is not difficult to obtain, by virtue of (4) and (12), for all \(P\) in \(A^*\), except for a finite number of points, the inequality

\[ R(t_0,\bar x(t_0),\bar u(t_0))\ge R(t_0,\widetilde x(t_0),\widetilde u(t_0)), \]

which holds for any pair $\tilde{x}, \tilde{u} \in E$, equality being attained only when $\tilde{x}(t)=\bar{x}(t)$ and $\tilde{u}(t)=\bar{u}(t)$. Since the choice of the point $P$ in $A^*$ is arbitrary, the inequality just written is equivalent to (7). Taking into account, further, that $\bar{x}, \bar{u}\in D$, together with relation (7) we shall have

\[ R\bigl(t,\bar{x}(t),\bar{u}(t)\bigr)=\sup_{x,u\in D} R(t,x,u). \]

Then from (6) we obtain that for all $\tau\in S$ relation (8) holds.

For simplicity of reasoning we assumed that an absolute minimum in the class $D$ exists. If this assumption is not made, the following more general result can be formulated.

Theorem 1*. Let there be a sequence of pairs $\{x_s(t),u_s(t)\}\subset D$. In order that this sequence be minimizing for the functional (1) on the set $D$, it is necessary and sufficient that there exist functions $\varphi_j(t,x)$ $(j=1,\ldots,m)$ such that:

1) for all $t\in A^*$, except for a finite number of points,

\[ \lim_{s\to\infty} R\bigl(t,x_s(t),u_s(t)\bigr)=r(t),\qquad r(t)=\sup_{x,u\in E} R(t,x,u); \]

2) for all $\tau\in S$

\[ \lim_{s\to\infty} G\bigl(x_s(\tau)\bigr)=g(\tau),\qquad g(\tau)=\inf_{x(\tau)\in B(\tau)} G\bigl(x(\tau)\bigr). \]

If the functional $F(x(\tau))$ contains one fixed $x=x^*(\tau)$ or, in other words, the functional being minimized has the form

\[ I(x,u)=\int_A f^0(t,x,u)\,dt+F^*(\tau), \tag{13} \]

then from Theorem 1 it follows

Theorem 2. Let there be a pair $\bar{x}(t),\bar{u}(t)\in D$. In order that this pair minimize the functional (13) on the set $D$, it is necessary and sufficient that there exist functions $\varphi_j(t,x)$ $(j=1,\ldots,m)$ such that, for all $t\in A^*$, except for a finite number of points,

\[ R\bigl(t,\bar{x}(t),\bar{u}(t)\bigr)=\sup_{x,u\in E} R(t,x,u). \]

For the case of one independent variable $(m=1)$

\[ I(x,u)=\int_0^{t_1} f^0(t,x,u)\,dt+F\bigl(x(0),x(t_1)\bigr), \tag{14} \]

and a consequence of Theorem 1 will be

Theorem 3. Let there be a pair $\bar{x}(t),\bar{u}(t)\in D$. In order that this pair minimize the functional (14) on the set $D$, it is necessary and sufficient that there exist a function $\varphi(t,x)$ such that:

1) for all $t\in(0,t_1)$, except for a finite number of points,

\[ R\bigl(t,\bar{x}(t),\bar{u}(t)\bigr)=\sup_{x,u\in E} R(t,x,u); \tag{15} \]

2)

\[ G\bigl(\bar{x}(0),\bar{x}(t_1)\bigr)= \inf_{\substack{x(0)\in B(0),\,x(t_1)\in B(t_1)}} G\bigl(x(0),x(t_1)\bigr), \tag{16} \]

where

\[ G\bigl(x(0),x(t_1)\bigr)=F\bigl(x(0),x(t_1)\bigr)+\varphi\bigl(t_1,x(t_1)\bigr)-\varphi\bigl(0,x(0)\bigr). \]

Remark 1. The sufficiency of conditions (15), (16) for the absolute minimum of the functional (14) was established in \({}^{1}\). Moreover, Krotov in \({}^{1,2}\) put forward the intuitive conjecture that these conditions are at the same time necessary. The properties of the functions \(f^i(t,x,u)\) \((i=0,1,\ldots,n)\) indicated there differ somewhat from those adopted in the present note and are, evidently, insufficient for the validity of the conjecture stated.

Remark 2. Along with the case considered of a finite number of points at which \(u^k(t)\) cease to be continuous, one may assume the presence, in these functions, of singularities on a set of points of measure zero, on a finite number of lines, etc. In accordance with the assumptions indicated, condition (7) in Theorem 1 and the analogous conditions in the subsequent theorems must be satisfied for all \(t \in A^*\), except for the corresponding set of points (lines).

Moscow Civil Engineering Institute
named after V. V. Kuibyshev

Received
25 II 1965

REFERENCES

\({}^{1}\) V. F. Krotov, Automation and Remote Control, 23, No. 12, 1571 (1962). \({}^{2}\) V. F. Krotov, Automation and Remote Control, 24, No. 5, 581 (1963). \({}^{3}\) V. F. Krotov, Automation and Remote Control, 25, No. 7, 1037 (1964). \({}^{4}\) A. G. Butkovskii, Automation and Remote Control, 24, No. 3, 314 (1963).