UDC 519.35

MATHEMATICS

M. M. KHRUSTALEV

ON SUFFICIENT CONDITIONS FOR AN ABSOLUTE MINIMUM

(Presented by Academician L. S. Pontryagin, 9 VIII 1966)

Consider the problem of the absolute minimum of the functional

\[ I(y(t), u(t))=\int_{t_0}^{t_1} f^0(t,y,u)\,dt+F(y_0,y_1). \tag{1} \]

Here \(y_0=y(t_0)\), \(y_1=y(t_1)\); \(y(t)\) is a vector-function, continuous and piecewise differentiable on the interval \([t_0,t_1]\subset T\) (\(T\) is the real line), with values in the \(n\)-dimensional real vector space \(Y\). For all \(t\in [t_0,t_1]\), the vector \(t,y(t)\in B\), a given subset of the space \(T\times Y\). The set of such functions \(y(t)\) will be denoted by \(D_y\). The vector-function \(u(t)\) (\(r\)-dimensional) is defined on \([t_0,t_1]\) and is continuous everywhere on \([t_0,t_1]\), except for a finite number of points where it may have discontinuities of the first kind. For each \(t,y(t)\in B\), the vector \(u(t)\in Q(t,y)\), a given subset of the \(r\)-dimensional vector space \(U\).

The conditions imposed on \(y(t), u(t)\) define, for each fixed \(t\), sets: \(B(t)\), the admissible values of \(y\), and \(V(t)\), the admissible values of the pairs \(y,u\).

In addition to the conditions listed above, the pair of functions \(y(t), u(t)\) must satisfy the system of \(n\) differential equations

\[ \dot y=f(t,y,u) \tag{2} \]

and the boundary conditions:

\[ y_0\in B_0\subset B(t_0),\qquad y_1\in B_1\subset B(t_1). \]

The vector-function \(f(t,y,u)\) (\(n\)-dimensional) and the function \(f^0(t,y,u)\) are continuous in all arguments for \(t\in [t_0,t_1]\), \(y,u\in V(t)\); \(F(y_0,y_1)\) is a continuous function defined on the set \(B(t_0)\times B(t_1)\).

The totality of all pairs of vector-functions \(y(t), u(t)\) having the properties listed above will be denoted by \(D\).

The problem is formulated as follows. Find a sequence \(\{ \bar y_s(t), \bar u_s(t)\}\subset D\) such that the functional (1) on this sequence tends, as \(s\to\infty\), to its least value \((^1)\).

Analogous conditions were considered in the works \((^1,^2)\); however, in them stronger conditions are imposed on the function \(\varphi(t,y)\), which can be weakened. This is made possible by Lemma 1. On the other hand, the refusal to consider the problem directly in the class \(D\), as is done in \((^3,^4)\), makes it possible to broaden the range of problems under consideration and substantially weaken the a priori conditions introduced by the method of proof.

1. Basic lemma. Let sets \(M,N\) be given, and let a functional \(I(v)\), \(v\in M\), bounded below, be defined on the set \(M\):

\[ \inf_{v\in M} I(v)=m>-\infty. \]

It is possible to prove the following lemma, which is a generalization of the lemma of work \((^1)\).

Lemma 1. In order that the sequence \(\{\bar v_s\}\subset M\) minimize the functional \(I\) on the set \(M\), it is sufficient that there exist a functional \(L(v)\), \(v\in N\), such that:

1) \(I(\bar v_s)\to l_1\le l,\quad \bar v_s\in M,\quad l=\inf\limits_{v\in N} L(v);\)

2) For any element \(v\in M\) and any prescribed \(\varepsilon>0\) there exists an element \(v^*\in N\) such that \(L(v^*)-I(v)\le\varepsilon\).

2. Sufficient conditions for optimality. Let the set \(B\) be open and let \(y(t)\in D_y\); \(y^*(t)=y(t)+\Delta y_0\), where \(\Delta y_0\in Y\) is a constant vector. Let, further, \(P\subset B\) be a set of measure \(0\) in \(B\), closed in \(B\).

Lemma 2. For any neighborhood \(W_0\subset B(t_0)\) of the point \(y(t_0)\) there exists a vector \(\Delta y_0\) such that \(y_0^*\in W_0\), the curve \(t,y^*(t)\in B\), and it intersects the set \(P\) on a set of measure \(0\) on the interval \([t_0,t_1]\).

The proof is analogous to that given in (3) (pp. 491—493), but, in contrast to (3), the “shifted” curve is not a solution of equations (2).

Definition 1. We shall call a set \(Q(x)\subset U\) continuous with respect to \(x\), \(x\in B\), if for any \(u_1\in Q(x)\), \(u_2\in Q(x+\Delta x)\) the following condition is fulfilled: for arbitrary \(\varepsilon>0\) there is a \(\delta(x)>0\) such that, if \(\|\Delta x\|<\delta(x)\), \(x+\Delta x\in B\), then there exist \(u_1'\in Q(x+\Delta x)\), \(u_2'\in Q(x)\) such that \(\|u_1-u_1'\|<\varepsilon,\ \|u_2-u_2'\|<\varepsilon\).

Definition 2. We shall say that a set \(Q(x)\subset U\), \(x\in B\), satisfies the Lipschitz condition with respect to \(x\), if for any \(u_1\in Q(x)\) and any \(\Delta x\), \(x+\Delta x\in B\), there exists \(u_2\in Q(x+\Delta x)\) such that \(\|u_1-u_2\|<\gamma\|\Delta x\|\), \(0\le\gamma<\infty\). Here \(\|\ \|\) is the Euclidean norm.

Definition 3. We shall say that a function \(\varphi(t)\) satisfies the one-sided Lipschitz condition if

\[ \varphi(t'')-\varphi(t')\le K(t''-t'),\qquad t''\ge t'. \]

Let the set \(Q(t,y)\) appearing in the formulation of the problem be nonempty for all \(t,y\in B\) and continuous with respect to \(t,y\) in the sense of Definition 1. Introduce into consideration a scalar function \(\varphi(t,y)\), continuous and continuously differentiable on \(B\setminus P\), and locally satisfying the Lipschitz condition in \(B\). Analogously to how this was done in (1), construct the functions

\[ \Phi(y_0,y_1)=F(y_0,y_1)+\varphi(t_1,y_1)-\varphi(t_0,y_0), \]

\[ R(t,y,u)=\varphi_y f(t,y,u)+\varphi_t-f^0(t,y,u). \]

Let the function \(\varphi(t,y)\) be chosen so that the function

\[ \mu(t)=\sup_{y,u} R(t,y,u),\qquad y\in B(t)\setminus P(t),\qquad u\in Q(t,y), \]

\[ P(t)=\{y:\ t,y\in P\}, \]

is summable and the number

\[ \bar\Phi=\inf_{y_0,y_1}\Phi(y_0,y_1),\qquad y_0\in B_0\setminus P(t_0),\qquad y_1\in B_1\setminus P(t_1). \]

is finite.

Consider the set \(E^*\) of pairs of vector functions \(y^*(t),u^*(t)\) such that the function \(y^*(t)\) differs from the functions \(y(t)\in D_y\) in that it may have discontinuities at the points \(t_0,t_1\) and intersects the set \(P\) on a set of measure \(0\) on the interval \([t_0,t_1]\), while the function \(u^*(t)\) is an arbitrary bounded function given on \([t_0,t_1]\), \(u^*(t)\in Q(t,y)\) (\(u^*(t)\) is not necessarily measurable).

On the set \(E^*\) define the functional

\[ L(y^*(t),u^*(t))=\Phi(y_0^*,y_1^*)-(\bar P)\int_{t_0}^{t_1}R(t,y^*(t),u^*(t))\,dt, \]

Here \((\overline{P})\ \displaystyle\int_{t_0}^{t_1}\) is the upper Perron integral (5). The following holds.

Lemma 3. For any pair \(y,u \in D\) and any \(\varepsilon>0\) there exists a pair \(y^*,u^* \in E^*\) such that \(L(y^*,u^*)-I(y,u)\leqslant \varepsilon\).

For the proof, one must take, as \(y^*(t)\), curves of the form appearing in Lemma 2 and consider the indicated difference.

On the basis of Lemmas 1 and 3 the following is proved.

Theorem. Let there be a sequence \(\{\overline{y}_s,\overline{u}_s\}\subset D\). In order that it minimize the functional \(I\) on \(D\), it is sufficient that there exist a function \(\varphi(t,y)\), continuous, locally satisfying in \(B\) the Lipschitz condition, continuously differentiable everywhere on \(B\setminus P\), such that

\[ I(\overline{y}_s,\overline{u}_s)\to \overline{\Phi}-\int_{t_0}^{t_1}\mu(t)\,dt,\qquad s\to\infty . \tag{3} \]

In the proof, as the set \(M\) one must take the set \(D\), and as \(N\) the set \(E^*\).

Remark 1. If the sequence \(\{\overline{y}_s,\overline{u}_s\}\subset D\cap E^*\) is such that \(|R(t,\overline{y}_s,\overline{u}_s)|\leqslant c(t)\), where \(c(t)\) is some summable function, then condition (3) may be replaced by the conditions \((^1)\)

\[ R(t,\overline{y}_s,\overline{u}_s)\to \mu(t)\quad \text{a.e.}, \]

\[ \Phi(\overline{y}_{0s},\overline{y}_{1s})\to \overline{\Phi},\qquad s\to\infty . \]

Remark 2. If \(f(t,y,u)\), \(f^0(t,y,u)\) are differentiable on \(B\) with respect to \(t,y\), and \(Q(t,y)=Q(t)\), i.e. does not depend on \(y\), then the conditions imposed on the function \(\varphi(t,y)\) in the theorem can be weakened.

Namely, \(\varphi(t,y)\) must be continuous and continuously differentiable on \(B\setminus P\), satisfy a one-sided Lipschitz condition along any solution \(y(t)\) of equation (2) (with piecewise-continuous \(u(t)\)) defined on the whole interval \([t_0,t_1]\) and passing through an arbitrary point of the set \(B(t_0)\), and also be bounded on each compact subset of \(B\). The function \(\varphi(t,y)\) in this case may be discontinuous.

Instead of Lemma 2 one must use the proof of Lemma 4 from \((^3)\); the proof of Lemma 3 is unchanged.

3. Estimate of the optimality of an arbitrary solution.
Let there be some function \(\varphi(t,y)\) satisfying the requirements indicated above. Let also there be an arbitrary pair \(\widetilde{y}(t),\widetilde{u}(t)\in D\) and \(\Phi(\widetilde{y}_0,\widetilde{y}_1)=\overline{\Phi}\). This pair satisfies the estimate

\[ \Delta(\widetilde{y}(t),\widetilde{u}(t))=I(\widetilde{y},\widetilde{u})-m \leqslant \varphi(t_0,\widetilde{y}_0)-\varphi(t_1,\widetilde{y}_1) +\int_{t_0}^{t_1}\left[\mu(t)+f^0(t,\widetilde{y},\widetilde{u})\right]\,dt . \]

The estimate is valid also in the case of a discontinuous \(\varphi(t,y)\). Similar estimates for the case of a smooth \(\varphi(t,y)\) were given by V. F. Krotov \((^6)\).

4. The case of an infinite interval of integration.
Let \(t_1=\infty\). In this case the formulation of the problem requires certain refinements. The functions \(y(t),u(t)\) must satisfy the conditions indicated above on any interval \([t_0,\tau]\subset [t_0,\infty)\).

Denote by \(A^\varepsilon\) the set of points \(y^\varepsilon:\{|y^\varepsilon-y|<\varepsilon,\ y\in A\subset Y,\ y^\varepsilon\in Y\}\). Then the expression \(y_1\in B_1\) is understood in the following sense. For any \(\varepsilon>0\) there exists \(\tau\), \(t_0\leqslant \tau<\infty\), such that \(y(t)\in B_1^\varepsilon\) for all \(t>\tau\).

The set \(D\) is defined basically exactly as before, but an admissible pair \(y(t),u(t)\) must satisfy the condition: \(I(y,u)\) is defined and finite.

To obtain sufficient conditions for optimality it is necessary to impose additional requirements on the functions \(f(t,y,u)\), \(f^0(t,y,u)\), \(\varphi(t,y)\) and on the set \(Q(t,y)\). Namely, they must be continuous and satisfy a Lipschitz condition (\(Q(t,y)\)—in the sense of Definition 2) with respect to \(t,y,u\) for all \(y(t)\in B^\varepsilon(t)\), \(\varepsilon=\mathrm{const}\), \(u\in Q(t,y)\), \(t\in [t_0,\tau]\), with a constant depending on \(t,y\) of the form \(\Omega(\xi)\), \(\xi=\|t,y\|^2\). The functions \(f(t,y,u)\) must not grow “too fast,” i.e. \(\|1,f\|\le \Omega(\xi)\).

Here the function \(\Omega(\xi)\) must be continuous, continuously differentiable, and such that the function \(\Omega'(\xi)/\Omega(\xi)\) is bounded, \(0\le \xi<\infty\). As such a function one may take, for example, the functions \(a+b\xi^q\), \(ae^{b\xi}\), \(q,a,b>0\), \(q\) an integer. The function \(\varphi_1=\lim_{t\to\infty}\varphi(t,y)\) must be continuous for all \(y_1\in B_1^\varepsilon\). The set \(P\) must have measure \(0\) in \(B^\varepsilon\), \(\varepsilon=\mathrm{const}\).

In this case the sufficient conditions are formulated in exactly the same way. In the proof of Lemma 2 it is necessary to take \(\Delta y\) not in the form \(\Delta y=\Delta y_0=\mathrm{const}\), but, for example, in the form \(\Delta y=\zeta e^{-t}\Delta y_0/\zeta+\Omega^3(\xi)\), \(\zeta=\mathrm{const}>0\).

5. Some other generalizations. The sufficient conditions are completely generalized if the functions \(f(t,y,u)\), \(f^0(t,y,u)\) have a discontinuity on some set \(P_1\) of measure \(0\) in \(B\), closed in \(B\), under the condition that all trajectories from \(D\) intersect \(P_1\) in a finite number of points, or under the condition that any trajectory not satisfying this requirement can be approximated in \(D\) (in the sense of convergence with respect to \(I\)) by trajectories of this kind.

The optimality conditions are also generalized to the case when \(t_1\) is not fixed, but the optimal value \(t_1\) is assumed finite. In this case one must require

\[ \overline{\Phi}=\inf_{t_1,y_0,y_1}\Phi(y_0,y_1,t_1), \qquad \mu(t)\equiv 0, \]

where \(t_1,\ y_1\in S\subset B\), \(S\) is given. This result for the case of smooth \(\varphi(t,y)\) was communicated to me by V. F. Krotov.

In a similar way the same results are obtained, under the same conditions except for the Lipschitz condition on \(\varphi(t,y)\) in the theorem, when instead of the set \(P\) one considers the piecewise-smooth set introduced in \((2)\).

In conclusion I express my sincere gratitude to V. F. Krotov for discussion of the paper and a number of valuable comments.

REFERENCES CITED

1. V. F. Krotov, Avtomatika i telemekh., 23, 12, 1571 (1962).
2. V. F. Krotov, Avtomatika i telemekh., 24, 5, 581 (1963).
3. V. G. Boltyanskii, Izv. AN SSSR, ser. matem., 28, 3, 481 (1964).
4. M. M. Baitman, DAN, 166, No. 2, 305 (1966).
5. E. Kamke, The Lebesgue–Stieltjes Integral, Moscow, 1959.
6. V. F. Krotov, Avtomatika i telemekh., 25, 11, 1521 (1964).