CYBERNETICS AND CONTROL THEORY

A. A. BLYUSIN

A PROBLEM OF A DISTRIBUTOR OPTIMIZING ON AN INTERVAL

(Presented by Academician L. S. Pontryagin, 29 II 1964)

In this note we consider certain problems on the optimal distribution of a given flow \(q(t)\) entering a certain system and distributed by it among \(n\) directions. Such a distribution of the flow is described mathematically by specifying \(n\) functions of time \(u_i(t) \geqslant 0\), \(i = 1, 2, \ldots, n\), subject to the condition: \(\sum u_i(t) = 1\). (\(u_i(t)\) expresses the fraction of the flow going to the \(i\)-th direction at time \(t\).) In the more general case, instead of the inequality \(u_i(t) \geqslant 0\) it may be required that the \(u_i(t)\) take values in certain prescribed intervals \([u_{i0}(t), u_{i1}(t)]\). We shall call a collection of functions \(u(t) = \{u_1(t), \ldots, u_n(t)\}\) a distributor* (although in the case where negative \(u_i(t)\) are allowed, this name is rather conventional). Suppose it is required to find such a distributor that would minimize the functional \(\Phi_u(T) = \sum f_i(T, x_i(T))\), where the \(f_i(t, x_i)\) are differentiable in \(t\) and in \(x_i\) almost everywhere,

\[ x_i(t) = \int_0^t q(\tau) u_i(\tau)\,d\tau, \]

and \(x_i(t)\) may take values in certain prescribed intervals \([x_{i0}(t), x_{i1}(t)]\).

The solution of this problem, generally speaking, is nonunique. This may be interpreted in such a way that additional conditions can be imposed on the desired optimal distributor \(u^*(t)\). In some problems of flow distribution it is natural to impose the requirement that one and the same distributor \(u^*(t)\) be optimal in the minimization of each functional \(\Phi_u(T)\), where \(T \in [T_0, T_1]\). Such a distributor will be called optimizing on the interval \([T_0, T_1]\). Analogously, one can define a distributor optimizing on a certain set** of values of \(t\).

Let us consider the space of phase trajectories \(\{x_i(t)\}\). Then it can be proved that, if \(u^*(t)\) optimizes on the interval \([T_0, T_1]\), there exist functions \(\lambda(T)\) and \(\alpha_i(T)\), \(0 \leqslant \alpha_i(T) \leqslant 1\), such that at the points of the corresponding phase trajectory \(\{x_i^*(T)\}\) the following conditions are satisfied:

\[ \frac{\partial f_i^-}{\partial x_i} + \alpha_i \left( \frac{\partial f_i^+}{\partial x_i} - \frac{\partial f_i^-}{\partial x_i} \right) = \lambda(T), \qquad X_{i0}(T) < x_i^*(T) < X_{i1}(T); \tag{1} \]

\[ \frac{\partial f_i^+}{\partial x_i} \geqslant \lambda(T), \qquad x_i^*(T) = X_{i0}(T); \tag{2} \]

\[ \frac{\partial f_i^-}{\partial x_i} \leqslant \lambda(T), \qquad x_i^*(T) = X_{i1}(T), \tag{3} \]

where

\[ f_i^+ = f_i(T, x_i^* + 0), \qquad f_i^- = f_i(T, x_i^* - 0), \]

\[ X_{i0}(T) = \max \left\{ x_{i0}(T), \int_0^T q(\tau)\,[U_{i0}(\tau)\theta(q) + U_{i1}(\tau)\theta(-q)]\,d\tau \right\}; \]

* Here and below \(i = 1, \ldots, n\).

** In what follows, unless otherwise specified, we shall understand \(T \in [T_0, T_1]\).

\[ X_{i1}(T)=\min\left\{x_{i1}(T)\int_0^T q(\tau)\left[U_{i0}(\tau)\theta(-q)+U_{i1}(\tau)\theta(q)\right]d\tau\right\}; \]

\[ U_{i0}(t)=\max\left\{u_{i0}(t),\ 1-\sum_{j\ne i}u_{j1}(t)\right\}; \]

\[ U_{i1}(t)=\max\left\{u_{i1}(t),\ 1-\sum_{j\ne i}u_{j0}(t)\right\}; \qquad \theta(q)= \begin{cases} 1, & q(t)>0,\\ 1, & q(t)\le 0. \end{cases} \]

If \(\partial^2 f_i/\partial x_i^2>0,\ x_i(T)\in[X_{i0}(T),\,X_{i1}(T)]\), then the listed conditions are also sufficient, and the distributor \(u^*(T)\) is unique. If, in addition, \(\partial^2 f_i/\partial x_i\partial t\equiv0,\ u_{i0}\equiv0,\ u_{i1}\equiv1,\ \dot x_{i0}\equiv \dot x_{i1}\equiv0\), then one can assert that an optimizing distributor exists everywhere.*

Solving the system of ordinary differential equations

\[ \dot x_i(t)= \begin{cases} h_i\left(q-\displaystyle\sum_{N_0(t)}\dot X_{j0}-\displaystyle\sum_{N_1(t)}\dot X_{j1} +\displaystyle\sum_{N(t)}h_j\frac{\partial^2(f_j-f_i)}{\partial x_j\partial t}\right) \left(\displaystyle\sum_{N(t)}h_j\right)^{-1}, & i\in N(t),\\[1.2em] \dot X_{i0}, & i\in N_0(t),\\ \dot X_{i1}, & i\in N_1(t),\\ 0, & i\in M(t), \end{cases} \]

where **

\[ N(t)=\left\{j:\frac{\partial f_j^-}{\partial x_j}=\lambda(t)\ \text{or}\ \frac{\partial f_j^+}{\partial x_j}=\lambda(t)\right\},\qquad M(t)=\{j:X_{j0}<x_j(t)<X_{j1}(t),\ 0<\alpha_j(t)<1\}, \]

\[ N_0(t)=\left\{j:\frac{\partial f_j^+}{\partial x_j}>\lambda(t),\ x_j(t)=X_{j0}(t)\right\},\qquad N_1(t)=\left\{j:\frac{\partial f_j^-}{\partial x_j}<\lambda(t),\ x_j(t)=X_{j1}(t)\right\}, \]

\[ h_j=\left(\frac{\partial^2 f_j}{\partial x_j^2}\right)^{-1}, \]

with the initial conditions \(x_i(0)=0\), we obtain a certain phase trajectory \(\{x_i^{**}(t)\}\). It has the property that, if

\[ q(t_0)u_{i0}(t_0)\le \dot x_i^{**}(t_0)\le q(t_0)u_{i1}(t_0), \tag{4} \]

then \(t_0\) may be regarded as \(T_0\) up to \(t_1\), when (4) ceases to hold, and \([q(t)]^{-1}\dot x_i^{**}(t),\ q(t)\ne0,\ t_0\le t\le t_1\), as \(u^*(T)\), \(T\in[t_0,t_1]\). With its help one can also obtain the optimal distributor \(u^*(t)\), \(0\le t<T_0\), satisfying the condition \(\Phi_{u^*}(T_0)=\min_{u(t)}\Phi_u(T_0)\).

When there is a mutually one-to-one dependence

\[ s=\int_0^t q(\tau)\,d\tau \]

it is not difficult to obtain the corresponding distributor \(v^*(s)=u^*(t)\), regarding \(s\) as the new independent variable. In general, however, in this case it is easier to obtain the optimizing distributor \(v^*(s)\) directly in parametric form, usually taking the quantity \(\lambda\) as the parameter.

In the case when, for example, \(\partial f_i/\partial t\equiv0,\ \dot f_i>0,\ u_{i0}=x_{i0}=0,\ u_{i1}=x_{i1}=\infty\), the everywhere-optimizing distributor (the unique

* Everything said above remains valid also in the case where, instead of \(f_i(t,x_i)\), one considers \(f(t,x_1,\ldots,x_n)\). In this case conditions (1)—(3) will be necessary and sufficient if

\[ \sum_{i,j=1}^{n}\frac{\partial^2 f}{\partial x_i\partial x_j}\xi_i\xi_j>0,\qquad \xi_i,\xi_j\ne0. \]

** If \(j\in N\), then the derivatives in the equations are taken from the right or from the left depending on the side from which the corresponding equality is satisfied.

everywhere) is obtained simply:

\[ v_i^*(\lambda)= \begin{cases} \dot\varphi_i(\lambda)\left[\displaystyle\sum_{N(\lambda)} \dot\varphi_j(\lambda)\right]^{-1}, & i\in N(\lambda),\\[6pt] 0, & i\in \overline{N}(\lambda), \end{cases} \]

\[ s(\lambda)=\sum_{N(\lambda)+M(\lambda)} \varphi_j(\lambda), \]

where \(\varphi_i(\lambda)\) is the real root of equation (1) for fixed \(\lambda\). When the values of \(\lambda\) run from \(\min_i \dot f_i(0)\) to \(\min_i \lim_{x_i\to\infty} f(x_i)\), then \(s\) varies from 0 to \(\infty\). In this case conditions (1)—(3), obviously, are satisfied with observance of the constraints imposed on the distributor and on the phase coordinates.

In particular, when \(f_i(x_i)=c_i\exp(-r_i^{-1}x_i)\), where \(c_i,r_i\) are positive constants and the numbering is chosen so that \(c_i/r_i \geq c_{i+1}/r_{i+1}\), then the optimizing distributor everywhere has the form:

\[ v_i^*(k)= \begin{cases} \dfrac{r_i}{R_k}, & 1\leq i\leq k,\\[6pt] 0, & i>k, \end{cases} \]

\[ \sum_1^{k-1}\Delta_m \leq s(k) < \sum_1^k \Delta_m, \]

where

\[ \Delta_m=R_m\ln\frac{c_i r_{i+1}}{c_{i+1}r_i},\qquad R_m=\sum_1^m r_i. \]

Here, in the role of the parameter it was more convenient to take the number of the direction for which the condition

\[ \dot f_k(0)=\max_j \dot f_j(0),\qquad j\in N(\lambda). \]

is satisfied. With the aid of this same parameter one can also obtain the dependence \(\Phi_{v^*}(s)\):

\[ \Phi_{v^*}(k)= R_k\exp\left[\frac{1}{R_k} \left(\sum_1^k r_i\ln\frac{c_i}{r_i}-s(k)\right)\right]. \]

In conclusion we make the following further remarks on certain properties of the distributors \(u^*(t)\) optimizing on the interval \([T_0,T_1]\), or of the corresponding distributors \(v^*(s)\).

Remark 1.

\[ \int_{T_0}^{T} F(\tau,\Phi_{u^*}(\tau))\,d\tau = \min_{u(t)} \int_{T_0}^{T} F(\tau,\Phi_u(\tau))\,d\tau, \qquad 0\leq t\leq T, \]

where \(\partial F(T,z)/\partial z \geq 0\). Hence, in particular, considering \(F(t,z)\equiv z\) and taking into account that for \(u_{i0}(t)\equiv-\infty,\ u_{i1}(t)\equiv\infty\) there exists everywhere an optimizing distributor, one may assert:

\[ \bar u^*(t)=\{u_i^*=x_i^*(t)[s(t)]^{-1}\} \]

is everywhere optimizing for

\[ \Phi_{\bar u}(T)= \sum_i\int_0^T \bar f_i(\tau,\bar u_i(\tau))\,d\tau, \]

where

\[ \bar f_i(t,z)=f_i(t,s(t)z), \]

and the constraints imposed on \(\bar u_i(t)\) are

\[ \sum_i \bar u_i(t)=1,\qquad x_{i0}(t)[s(t)]^{-1}\leq \bar u_i(t)\leq x_{i1}(t)[s(t)]^{-1}. \]

Remark 2. If \(u_{i0}(t), x_{i0}(t), \partial f_i/\partial t \equiv 0,\ u_{i1}(t), x_{i1}(t)\equiv\infty,\ -\dot f_i,\ddot f_i,\dot f_i,\ -\partial Q(t,z)/\partial z>0\), then the functional with “feedback”

\[ \hat\Phi_u(T)= \sum_i f_i\left(\int_0^T Q(\tau,\hat\Phi_u)u_i\,d\tau\right) \]

is minimized everywhere by the distributor \(v^*(s),\ 0\leq s<\infty\). Moreover, the latter also possesses the property of optimality with respect to speed of response: it ensures the equality \(\hat\Phi_u(T)=c\), where

\(c < \sum f_i(0)\), in minimum time. If, however, \(\partial Q(t,z)/\partial z > 0\), \(\dot f_i > 0\), and \(c > \sum f_i(0)\), then \(v^*(s)\) ensures optimality with respect to “duration of action.” Under the conditions considered, \(v^*(s)\) is unique in every sense.

Remark 3. If among the \(f_i(t,x_i)\) there is an \(f_{i_0}(t,x_{i_0})\) and \(u_{i_0 0} \equiv x_{i_0 0} \equiv -\infty\), then the distributor optimizing on \([T_0,T_1]\) is correspondingly also optimizing under the condition \(\sum u_i(t) \geqslant 1\) instead of \(\sum u_i(t)=1\).

For \(u_{i_0 1}=x_{i_0 1}=\infty\), the same is true for \(\sum u_i(t) \leqslant 1\). With a combination of the conditions, evidently, no restrictions need be imposed on \(\sum u_i(t)\).

Received
13 II 1964