On a Problem of Optimal Control Connected with Minimization of a Functional of the Type “Maximum Deviation”

B. G. Pittel

A problem is considered of minimizing a functional of the type “maximum deviation” for a linear system of differential equations with controllable coefficients. A method is proposed that reduces this problem to a sequence of optimal-control problems with integral functionals. As an illustration of the approach presented, the case of a scalar differential equation of order \(n\) with the functional “maximum modulus of the solution” is considered. A number of theorems connected with the structure of the optimal control are proved.

Introduction. Let a system be given

\[ \dot{x}=\left(A+\sum_{i=1}^{r} u_i B_i\right)x,\qquad x(0)=x_0, \tag{0.1} \]

where \(A, B_i\) are constant matrices of order \(n\times n\), \(x\) is an \(n\)-dimensional vector of phase variables, \(u=(u_1,\ldots,u_r)\) is a vector of control functions, and \(u\), for all \(t\), belongs to a prescribed bounded, closed, convex domain \(U\) of the space \(E_r\).

In a number of cases it is natural to evaluate the quality of control by functionals of the form

\[ J(u)=\sup_{0\leq t<+\infty}\varphi(x(t)),\qquad (\varphi(x)\geq 0,\ \varphi(0)=0). \tag{0.2} \]

The problem consists in choosing a control \(u(t)\) from the class of admissible ones that minimizes the functional (0.2). The physical meaning of the problem posed is evident: \(\varphi(x)\) is a measure of the deviation of the point \(x\) of phase space from the origin; it is required to choose a control minimizing the maximum value of this measure.

Since the functional (0.2) is not an integral one and is not reducible to one, the necessary optimality condition—the Pontryagin maximum principle [1]—is not directly applicable to the problem posed. In the present paper a natural method is proposed for reducing the problem posed to a sequence of problems with integral functionals. This method is analogous to the method applied in [2] in the study of the optimal time problem for approximating nonintegral constraints on the control by integral ones. We also note the works [3], [4], in which, using methods of functional analysis, necessary optimality conditions were obtained for functionals of the type (0.2) and for a finite time interval.

1. METHOD OF REDUCING THE PROBLEM TO A SEQUENCE OF PROBLEMS WITH INTEGRAL FUNCTIONALS

We shall assume that in (0.2) \(\varphi(x)\) is continuously differentiable. Consider the sequence of functionals

\[ J_p(u)=\left[\int_0^{+\infty}\rho(t)\varphi^p(x(t))\,dt\right]^{1/p}; \tag{1.1} \]

here \(\rho(t)\) is continuously differentiable, \(\rho(t)>0\), and

\[ \int_0^{+\infty}\rho(t)\,dt=1. \tag{1.2} \]

Denote by \(x_p(t)\), \(u^p(t)=(u_1^p(t),\ldots,u_r^p(t))\), respectively, the trajectory and the control optimal in the sense of the functional \(J_p(u)\), \((x_p(0)=x_0)\). (The existence of a solution \(x_p(t), u^p(t)\) follows from the obvious estimate

\[ \left(\int_0^{+\infty}\rho(t)\varphi^p(x(t))\,dt\right)^{1/p} \leq \sup_{0\leq t<+\infty}\varphi(x(t)), \]

from the assumption that there exists a solution optimal in the sense of the functional (0.2), and is proved in exactly the same way as the existence theorem in [5], part 1.) It is clear that \(x_p, u^p\) satisfy Pontryagin’s maximum principle and Bellman’s principle of optimality with respect to the functional \((J_p)^p\). Let \(p_i\) be a subsequence such that \(\lim_{i\to+\infty}p_i=+\infty\), and let \(x_{p_i}(t)\) and \(u^{p_i}(t)\) on any finite interval \([0,T]\) converge, respectively, uniformly to \(x_{+\infty}(t)\) and weakly in \(L_2[0,T]\) to \(u^{+\infty}(t)\) (see [5], part 1). We shall show that \(x_{+\infty}(t), u^{+\infty}(t)\) are optimal in the sense of the functional (0.2). Indeed, \(u^{+\infty}(t)\) is an admissible control (by the properties of the control domain \(U\)), and \(x_{+\infty}(t)\) is a solution of system (0.1) with \(u=u^{+\infty}(t)\), \(x_{+\infty}(0)=x_0\). Let \(u_1(t)\) be any admissible control, and let \(x_1(t)\) be the corresponding trajectory. Then, by the definition of \(x_{p_i}(t)\),

\[ \left(\int_0^{+\infty}\rho(t)\varphi^{p_i}(x_{p_i}(t))\,dt\right)^{1/p_i} \leq \left(\int_0^{+\infty}\rho(t)\varphi^{p_i}(x_1(t))\,dt\right)^{1/p_i} \leq \sup_{0\leq t<+\infty}\varphi(x_1(t)). \tag{1.3} \]

On the other hand, since for any finite \(T\), \(x_{p_i}(t)\) converge uniformly on \([0,T]\) to \(x_{+\infty}(t)\), and (see, for example, [6]),

\[ \lim_{p_i\to+\infty} \left(\int_0^T\rho(t)\varphi^{p_i}(x_{+\infty}(t))\,dt\right)^{1/p_i} = \max_{0\leq t\leq T}\varphi(x_{+\infty}(t)), \]

it follows that

\[ \max_{0\leq t\leq T}\varphi(x_{+\infty}(t)) = \lim_{p_i\to+\infty} \left(\int_0^T\rho(t)\varphi^{p_i}(x_{p_i}(t))\,dt\right)^{1/p_i} \leq \sup_{0\leq t<+\infty}\varphi(x_1(t)). \tag{1.4} \]

Hence, for any admissible trajectory \(x_1(t)\),

\[ \sup_{0\leq t<+\infty}\varphi(x_{+\infty}(t)) \leq \sup_{0\leq t<+\infty}\varphi(x_1(t)), \]

which was required to be proved.

Further, since in (1.3) \(x_1(t)\) may be replaced by \(x_{+\infty}(t)\), it follows from inequalities (1.3), (1.4) that

\[ \lim_{p_i\to+\infty} J_{p_i}(u^{p_i}) = \sup_{0\leq t<+\infty}\varphi(x_{+\infty}(t)) = J(u^{+\infty}). \tag{1.5} \]

But the numerical sequence \(J_p(u^p)\) is increasing. Indeed, applying Hölder’s inequality, we obtain: if \(p_1<p_2\), then

\[ J_{p_1}(u^{p_1}) = \left(\int_0^{+\infty}\rho(t)\varphi^{p_1}(x_{p_1}(t))\,dt\right)^{1/p_1} \leq \left(\int_0^{+\infty}\rho(t)\varphi^{p_1}(x_{p_2}(t))\,dt\right)^{1/p_1} \leq \]

\[ \leq \left(\int_0^{+\infty}\rho(t)\varphi^{p_2}(x_{p_2}(t))\,dt\right)^{1/p_2} = J_{p_2}(u^{p_2}). \]

Therefore the equality

\[ \lim_{p\to+\infty} J_p(u^p) = \sup_{0\leq t<+\infty}\varphi(x_{+\infty}(t)) = J(u^{+\infty}) \tag{1.6} \]

is valid.

Let us introduce into consideration the functions of the phase variables \(V(x)\), \(V_p(x)\), defining them by the formula

\[ V(x)=\min_{u\in U}J(u)=\min_{u\in U}\sup_{0\leq t<+\infty}\varphi(x(t)),\qquad x(0)=x, \]

\[ V_p(x)=\min_{u\in U}J_p(u)=\min_{u\in U}\left(\int_0^{+\infty}\rho(t)\varphi^p(x(t))\,dt\right)^{1/p},\qquad x(0)=x. \tag{1.7} \]

Then relation (1.6) takes the form

\[ V(x)=\lim_{p\to+\infty}V_p(x). \tag{1.8} \]

It must be noted that trajectories optimal in the sense of functional (0.2), because of its specific nature, do not, generally speaking, satisfy Bellman’s optimality principle. However, for the trajectories \(x_{+\infty}(t)\) obtained by the limiting passage indicated above, one can in a number of cases establish the validity of the optimality principle by using the following theorem.

Theorem 1. The segment of the trajectory \(x_{+\infty}(t)\), beginning at the point \(x_{+\infty}(t_0)\), is optimal in the sense of the functional \(J(u)=\sup_{t\geq t_0}\varphi(x(t))\) for all those \(t_0\), \(0\leq t_0<+\infty\), for which the points \(x_{+\infty}(t_0)\) are points of continuity of the function \(V(x)\).

Proof. We shall prove by contradiction. Suppose that there exists a point \(t_0\in[0,+\infty)\) such that \(x_{+\infty}(t_0)\) is a point of continuity of the function \(V(x)\), while the trajectory \(x_{+\infty}(t)\) on the interval \([t_0,+\infty)\) is not optimal in the sense of functional (0.2). This means that

\[ \sup_{t\geq t_0}\varphi(x_{+\infty}(t))>V(x_{+\infty}(t_0)). \tag{1.9} \]

Denote by \(x_{p_i}^*(t)\) a trajectory optimal in the sense of functional (0.2), \(x_{p_i}^*(t_0)=x_{p_i}(t_0)\). Since the trajectories \(x_{p_i}(t)\) satisfy the optimality principle for the functionals \(J_p(u)\), the following sequence of inequalities holds:

\[ \sup_{t_0\leq t<+\infty}\varphi(x_{+\infty}(t)) = \lim_{T\to+\infty}\lim_{p_i\to+\infty} \left(\int_{t_0}^{T}\rho(t)\varphi^{p_i}(x_{p_i}(t))\,dt\right)^{1/p_i} \leq \]

\[ \leq \lim_{p_i \to +\infty} \left( \int_{t_0}^{+\infty} \rho(t)\varphi^{p_i}(x_{p_i}(t))\,dt \right)^{1/p_i} \leq \lim_{p_i \to +\infty} \left( \int_{t_0}^{+\infty} \rho(t)\varphi^{p_i}(x_{p_i}^{*}(t))\,dt \right)^{1/p_i} \leq \]

\[ \leq \lim_{p_i \to +\infty}\sup_{t_0\leq t<+\infty}\varphi(x_{p_i}^{*}(t)) = \lim_{p_i \to +\infty} V(x_{p_i}(t_0)) = V(x_{+\infty}(t_0)), \]

i.e.

\[ \sup_{t\geq t_0}\varphi(x_{+\infty}(t))\leq V(x_{+\infty}(t_0)), \]

which contradicts (1.9). The theorem is proved.

Remarks. 1. It seems that the condition of the theorem can be weakened by requiring only upper semicontinuity of the function \(V(x)\) at the point \(x_{+\infty}(t_0)\), i.e., by assuming that

\[ \overline{\lim}_{x\to x_{+\infty}(t_0)} V(x)\leq V(x_{+\infty}(t_0)). \tag{1.10} \]

However, in fact condition (1.10) is equivalent to continuity of \(V(x)\) at the point \(x_{+\infty}(t_0)\), since it can be shown that the function \(V(x)\) is lower semicontinuous at any point \(x_0\), i.e.,

\[ \underline{\lim}_{x\to x_0} V(x)\geq V(x_0). \]

1. The function \(V(x)\) is continuous everywhere for a finite time interval, i.e., in the case when

\[ V(x)=\min_{u\in U} J(u)=\min_{u\in U}\max_{0\leq t\leq T}\varphi(x(t)),\qquad x(0)=x. \]

Therefore, in this case the optimal trajectories \(x_{+\infty}(t)\) satisfy Bellman’s principle of optimality at all their points.

1. Everything stated is also valid for the more general system of equations of the form

\[ \dot{x}=\left(A+\sum_{i=1}^{r}u_iB_i\right)x+Cu, \]

where \(C\) is a rectangular matrix of order \(n\times r\).

2. THE CASE OF A SCALAR EQUATION OF \(n\)-TH ORDER

Let the motion be described by an equation of the form

\[ x^{(n)}+u_n x^{(n-1)}+\ldots+u_1x=0, \tag{2.1} \]

\[ m_i\leq u_i\leq M_i,\qquad 1\leq i\leq n;\qquad x^{(j)}(0)=x_0^{(j)},\qquad 0\leq j\leq n-1. \]

We shall evaluate the quality of control by the value of the functional

\[ J(u)=\sup_{0\leq t<+\infty}|x(t)|. \tag{2.2} \]

The expediency of estimating the maximum deviation of the trajectory in \(n\)-dimensional phase space from the origin by the functional (2.2) is connected with its simplicity and with the inequality (see [5], part 2):

\[ \sup_{0\leq t<+\infty}\max_{0\leq k\leq n-1}|x^{(k)}(t)| \leq C\sup_{0\leq t<+\infty}|x(t)|; \]

here the constant \(C\) is common for all initial conditions and for all admissible controls.

In order to use the results of the preceding section, let us introduce the functionals

\[ J_p(u)=\left(\int_0^{+\infty}\rho(t)|x(t)|^p\,dt\right)^{1/p}; \tag{2.3} \]

\[ \rho(t)>0 \text{ is continuously differentiable},\qquad \int_0^{+\infty}\rho(t)\,dt=1. \]

Let \(u^p(t)=(u_1^p(t),\ldots,u_n^p(t))\) be a control optimal in the sense of the functional \(J_p(u)\). Repeating, with minor changes, the proof of the theorem* in [7], one can prove the following theorem.

Theorem 2 (On the structure of the optimal control \(u^p(t)\)). Suppose that in (2.1), for \(n\geqslant 3\),

\[ M_1\cdot m_1\ne 0. \tag{2.4} \]

Then, for any \(p\), at least \(\alpha_n\) of the functions \(u_1^p(t),\ldots,u_{\alpha_n}^p(t)\) are piecewise-constant functions of \(t\), assume only their extreme values, \(u_i^p(t)—m_i\) and \(M_i\), \(i=1,2,\ldots,\alpha_n\), and on any finite time interval have only a finite number of discontinuities. Here

\[ \alpha_1=1,\qquad \alpha_2=2,\qquad \alpha_n=\left[\frac{n}{2}\right]+2 \quad \text{for } n\geqslant 3. \]

The main result of the present section will be the proof of the following assertion.

Theorem 3. If condition (2.4) is fulfilled, then among the controls optimal in the sense of the functional (2.2) there exists one such that at least \(\beta_n\) of the functions \(u_1(t),\ldots,u_{\beta_n}(t)\) are piecewise-constant functions of \(t\), assuming only their extreme values, \(u_i(t)—m_i\) and \(M_i\), \(i=1,2,\ldots,\beta_n\), whose discontinuities can accumulate only at infinity. Here

\[ \beta_1=1,\qquad \beta_n=\left[\frac{n+3}{2}\right]\quad \text{for } n\geqslant 2. \]

We first prove an auxiliary assertion, which is a certain generalization of the Vallée-Poussin theorem (see [8]).

Lemma. On an arbitrary interval of length \(h_*\), any solutions of the equations

\[ \frac{d^k}{dt^k}\left(z^{(n-k)}+u_n z^{(n-k-1)}+\cdots+u_{k+1}z\right) +u_k z^{(k-1)}+\cdots+u_1z=0, \tag{2.5} \]

\[ m_i\leqslant u_i\leqslant M_i,\qquad 1\leqslant i\leqslant n,\qquad 0\leqslant k\leqslant \left[\frac{n+1}{2}\right] \]

have no more than \((n-1)\) zeros. Here

\[ h_*=\min_{0<k\leqslant \left[\frac{n+1}{2}\right]} h_k \tag{2.6} \]

and \(h_k\) is the positive root of the equation

\[ P_k(h)=Q_k(h); \]

\[ P_k(h)=1-\lambda_n\frac{h}{1!}-\cdots-\lambda_{k+1}\frac{h^{\,n-k}}{(n-k)!}; \]

* This theorem pertains to the problem of optimal speed.

\[ \lambda_i=\max\left(|m_i|,\ |M_i|\right), \qquad 1\leq i\leq n, \]

\[ Q_k(h)= \begin{cases} 0, & \text{for } k=0,\\[6pt] \dfrac{\lambda_1 h^n}{k!(n-k)!}+\cdots+ \dfrac{\lambda_k h^{\,n-k+1}}{k!(n+1-2k)!}, & \text{for } 0<k\leq \left[\dfrac{n+1}{2}\right]. \end{cases} \tag{2.7} \]

Proof of the lemma. Suppose the assertion of the lemma is false. Then, for some \(k_0\), \(0\leq k_0\leq \left[\dfrac{n+1}{2}\right]\), there exists a solution of (2.5) which on the interval \([t_0,t_0+h_*]\) has at least \(n\) zeros. According to (2.7),

\[ P_{k_0}(h)>0 \quad \text{for } 0\leq h<h_* . \]

Therefore, by the Vallée-Poussin theorem, any nonzero solution of the equation

\[ L_{k_0}(z)=z^{(n-k_0)}+u_n z^{(n-k_0-1)}+\cdots+u_{k_0+1}z=0 \tag{2.8} \]

has on the interval \([t_0,t_0+h_*]\) no more than \((n-k_0-1)\) zeros. Hence it follows (see [9]) that on the interval \([t_0,t_0+h_*]\) the operator \(L_{k_0}\) admits the representation

\[ L_{k_0}(z)=s_{n-k_0+1}(t)\frac{d}{dt}s_{n-k_0}(t)\cdots \frac{d}{dt}s_1(t)z, \]

\[ s_i(t)>0,\qquad t\in [t_0,t_0+h_*]. \]

Since the function \(z(t)\) has \(n\) zeros, by Rolle’s theorem and from the representation of the operator \(L_{k_0}\) we obtain that

\[ y(t)=z^{(n-k_0)}+u_n z^{(n-k_0-1)}+\cdots+u_{k_0+1}z \tag{2.9} \]

on the interval \([t_0,t_0+h_*]\) changes sign at least \(k\) times. Consequently, in particular, one of the numbers \(\lambda_1,\ldots,\lambda_{k_0-1},\lambda_{k_0}\) in (2.7) is nonzero, since otherwise equation (2.5) would take the form \(y^{(k)}=0\), and its solution would have no more than \((k-1)\) changes of sign. Therefore, according to the definition of \(h_*\),

\[ P_{k_0}(h_*)\geq Q_{k_0}(h_*)>0. \]

Denote

\[ \mu=\max_{t_0\leq t\leq t_0+h_*}\left|z^{(n-k_0-1)}(t)\right|,\qquad \nu=\max_{t_0\leq t\leq t_0+h_*}\left|y^{(k_0-1)}(t)\right|. \tag{2.10} \]

Using the lemma in [8] (Chap. 4, § 2), we shall have

\[ \int_{t_0}^{t_0+h_*} |z^{(j)}(t)|\,dt <\mu\,\frac{h_*^{\,n-k_0-j}}{(n-k_0-j)!}, \qquad \int_{t_0}^{t_0+h_*} |y(t)|\,dt <\nu\,\frac{h_*^{\,k_0}}{k_0!}, \tag{2.11} \]

\[ 0\leq j\leq n-k_0-1. \]

Integrating equality (2.9) from \(t_1\) to \(t_2\), where \(t_1\) is one of the zeros of \(z^{(n-k_0-1)}(t)\), and \(t_2\) is a point of maximum of the modulus of \(z^{(n-k_0-1)}(t)\), and applying (2.11), we obtain

\[ \mu< \frac{1}{P_{k_0}(h_*)} \int_{t_0}^{t_0+h_*}|y(t)|\,dt, \]

i.e.,
\[ \int_{t_0}^{t_0+h_*} |z^{(j)}(t)|\,dt < \frac{h_*^{\,n-k_0-j}}{P_{k_0}(h_*)(n-k_0-j)!} \int_{t_0}^{t_0+h_*} |y(t)|\,dt, \tag{2.12} \]
\[ 0 \leq j \leq n-k_0-1. \]

From (2.12) and (2.9) it follows, as is not difficult to show, that (2.12) is also valid for \(j=n-k_0\). Taking into account the second inequality (2.11), the estimates (2.12) take the form:
\[ \int_{t_0}^{t_0+h_*} |z^{(j)}(t)|\,dt < \frac{\nu h_*^{\,n-j}}{k_0!(n-k_0-j)!\,P_{k_0}(h_*)}, \tag{2.13} \]
\[ 0 \leq j \leq n-k_0. \]

Equation (2.5) can be written as follows:
\[ \frac{d^{k_0}y}{dt^{k_0}}+u_{k_0}z^{(k_0-1)}+\cdots+u_1z=0. \]

Integrating it over the interval from \(\tau_1\) to \(\tau_2\), where \(\tau_1\) is one of the zeros of \(y^{(k_0-1)}(t)\), and \(\tau_2\) is a point of maximum of the modulus \(y^{(k_0-1)}(t)\), \((\tau_1,\tau_2\subset [t_0,t_0+h_*])\), and taking into account (2.13) (recall that \(n-k_0\geq k_0+1\)), we finally arrive at the following inequality:
\[ \nu < \nu\,\frac{Q_{k_0}(h_*)}{P_{k_0}(h_*)}, \quad \text{or} \quad P_{k_0}(h_*)-Q_{k_0}(h_*)<0, \]
which is impossible according to the definition of \(h_*\).

The lemma is proved.

Proof of Theorem 3. The necessary condition for optimality of \(u^p(t)=(u_1^p(t),\ldots,u_n^p(t))\)—Pontryagin’s maximum principle—has in the present case the following form: there exists a nonzero absolutely continuous solution \(\psi=(\psi_1,\ldots,\psi_n),\psi_\tau,\psi_0\) of the system
\[ \begin{cases} \dot\psi_1=\psi_nu_1-p\psi_0\rho |x|^{p-1}\operatorname{sign}x, & \dot\psi_\tau=-\psi_0\dot\rho |x|^p,\\ \dot\psi_i=-\psi_{i-1}+\psi_nu_i,\quad i=2,\ldots,n; & \psi_0=\mathrm{const}\leq 0 \end{cases} \tag{2.14} \]
such that for almost all \(t\)
\[ H(u^p(t),x(t),t,\psi(t),\psi_\tau(t),\psi_0)= \]
\[ =\max_{u\in U} H(u,x(t),t,\psi(t),\psi_\tau(t),\psi_0)\equiv 0, \tag{2.15} \]
where
\[ H=\sum_{i=1}^{n-1}\psi_i(t)x^{(i)}(t) -\psi_n(t)\sum_{i=1}^{n}u_i x^{(i-1)}(t) +\psi_\tau(t)+\psi_0\rho |x|^p. \]

From the form of the function \(H\) and the control domain \(U\) it follows that
\[ u_i^p(t)= \begin{cases} m_i, & \text{if } \psi_n(t)x^{(i-1)}(t)>0,\\ M_i, & \text{if } \psi_n(t)x^{(i-1)}(t)<0. \end{cases} \tag{2.16} \]

The assertion of the theorem will therefore be proved if we show that the total number \(r_{\psi_n}(T)\) of zeros of the functions \(\psi_n(t), x(t),\ldots,x^{(n-1)}(t)\) on any

on a finite interval \([0,T]\) is finite and admits an estimate independent of \(p\), i.e., an estimate of the form

\[ r_{\beta_n}(T) \leq c(T). \tag{2.17} \]

Indeed, it is not difficult to show that in this case, for a subsequence of controls \(u^{p_i}(t)\) (see item 1), weakly convergent on any finite interval \([0,T]\) to \(u^{+\infty}(t)\), the switching instants of the functions \(u_1^p(t), \ldots, u_{\beta_n}^{p_i}(t)\) will have limits that are switching instants of the functions \(u_1^{+\infty}(t), \ldots, u_{\beta_n}^{+\infty}(t)\). At the same time, the total number of discontinuities of these functions on the interval \([0,T]\) does not exceed \(c(T)\) (see (2.17)).

Let us first estimate \(r_1(T)\)—the total number of zeros of the functions \(x(t), \psi_n(t)\) on the interval \([0,T]\). By the definition of the number \(h_*\) (see the lemma) and by the Vallée-Poussin theorem, the number of zeros of \(x(t)\) does not exceed \((n-1)m(T)\), where \(m(T)=\left[\dfrac{T}{h_*}\right]+1\). We shall show that on any interval of length \(h_*\) the number of zeros of the function \(\psi_n(t)\) does not exceed \((2n-1)\). Equation (2.1), the system for determining \(\psi\) in (2.14), can be written in vector-matrix notation:

\[ \dot y = Ay,\qquad \dot\psi = -A^* \psi - f(t)e_1, \]

where

\[ y = (x,\ x^{(1)},\ldots,\ x^{(n-1)}), \]

\[ e_1=(1,\ \underbrace{0,\ldots,0}_{n-1}), \]

\[ f(t)=\psi_0\rho(t)p|x(t)|^{p-1}\operatorname{sign}x(t), \]

\[ A= \begin{Vmatrix} 0 & 1 & 0 & \cdots & 0\\ \cdot & \cdot & \cdot & & \cdot\\ \cdot & & \cdot & & \cdot\\ \cdot & & 1 & & \cdot\\ \cdot & & & \cdot & 0\\ 0 & \cdot & \cdot & \cdot & \cdot & 1\\ -u_1 & \cdot & \cdot & \cdot & \cdot & -u_n \end{Vmatrix}. \]

According to what was stated above, on any interval \([t_0,t_0+h_*]\) the operator \(L(x)=x^{(n)}+\cdots+u_1x\) admits the representation

\[ L(x)=s_{n+1}(t)\frac{d}{dt}s_n(t)\frac{d}{dt}\cdots\frac{d}{dt}s_1(t)x, \]

\[ s_i(t)>0,\qquad s_{n+1}s_n\cdots s_1 \equiv 1,\qquad t\in[t_0,t_0+h_*]. \]

This means that in the variables \(z=(z_1,z_2,\ldots,z_n)\), defined by the formulas

\[ z_1=s_1x=b_{11}x, \]

\[ z_2=s_2\frac{d}{dt}s_1x=s_2\dot s_1x+s_2s_1\dot x =b_{21}x+b_{22}\dot x, \]

\[ \cdot\qquad \cdot\qquad \cdot\qquad \cdot\qquad \cdot\qquad \cdot\qquad \cdot\qquad \cdot\qquad \cdot \]

\[ z_n=s_n\frac{d}{dt}\cdots\frac{d}{dt}s_1x =s_n\dot s_{n-1}\cdots \dot s_1x+\cdots+s_ns_{n-1}\cdots s_1x^{(n-1)} \]

\[ = b_{n1}x+\cdots+b_{nn}x^{(n-1)}, \]

the system \(\dot y=Ay\) takes the form \(\dot z=A_0z\), where

\[ A_0=\dot B B^{-1}+BAB^{-1}= \begin{Vmatrix} 0&\dfrac{1}{s_2}&0&\ldots&0\\ .&.&.&.&.&.\\ .&.&.&.&.&0\\ .&.&.&.&\dfrac{1}{s_n}\\ 0&&.&.&.&0 \end{Vmatrix}; \]

\[ B= \begin{Vmatrix} b_{11}&&&O\\ b_{21}&b_{22}&&\\ .&.&.&\\ .&.&.&\\ b_{n1}&.&.&b_{nn} \end{Vmatrix}. \]

Then, as is not difficult to verify, the system
\(\dot\psi=-A^*\psi-f(t)e_1\) in the variables
\(\chi=(\chi_1,\ldots,\chi_n)\), defined by the formula
\(\chi=(B^*)^{-1}\psi\), is written as

\[ \dot\chi=-A_0^*\chi-f(t)(B^*)^{-1}e_1=-A_0^*\chi-f(t)/s_1(t)e_1, \]

or, eliminating the variables \(\chi_1,\ldots,\chi_{n-1}\),

\[ (-1)^n s_1(t)\frac{d}{dt}\ldots \frac{d}{dt}s_n(t)\frac{d\chi_n}{dt} =f(t)=\psi_0\rho(t)\rho |x|^{p-1}\operatorname{sign}x. \]

But by virtue of the formula \(\chi=(B^*)^{-1}\psi\) and the form of the matrix \(B\), \(\chi_n=s_{n+1}\psi_n\). Finally, for \(\psi_n\) we obtain the equation

\[ (-1)^n s_1(t)\frac{d}{dt}\ldots \frac{d}{dt}s_{n+1}\psi_n =\psi_0\rho(t)\rho |x|^{p-1}\operatorname{sign}x. \]

The function standing on the right-hand side of the equation obtained, for
\(t\in[t_0;\,t_0+h_*]\), changes sign no more than \((n-1)\) times. Hence it follows that on this interval the function \(\psi_n(t)\) has at most \((2n-1)\) zeros. From what was said above it then follows that

\[ r_1(T)\le (3n-2)m(T),\qquad m(T)=\left[\frac{T}{h_*}\right]+1. \tag{2.18} \]

Let the total number of zeros \(r_k(T)\) on the interval \([0,T]\) of the functions
\(\psi_n(t)\), \(x(t),\ldots,x^{(k-1)}(t)\),
\[ \left(1\le k\le \left[\frac{n+1}{2}\right]\right) \]
not exceed \(f_k(n)m(T)\). (It has already been proved that one may take
\(f_1(n)=(3n-2)\).) Then the number of intervals on the segment \([0,T]\) on each of which none of the functions
\(x(t),\ldots,x^{(k-1)}(t),\ \psi_n(t)\) changes sign is no more than
\(f_k(n)m(T)+1\). Consider one of these intervals \(a_s\le t\le b_s\).
According to (2.16), \(u_i^p(t)\equiv\mathrm{const}\), i.e., either
\(u_i^p\equiv m_i\), or \(u_i^p\equiv M_i\), for \(1\le i\le k\) and
\(t\in[a_s,b_s]\). Further, for these \(t\), \(x^{(k)}(t)\ne0\), since otherwise, by equation (2.1), for all \(t\in[a_s,b_s]\) the identity would hold:

\[ u_k x^{(k-1)}+\cdots+u_1 x=0, \]

\[ u_{k-1}x^{(k-1)}+\cdots+u_1 x^{(1)}=0, \]

\[ \cdots\cdots\cdots\cdots\cdots \]

\[ u_1 x^{(k-1)}=0, \]

whence \(x(t)\equiv 0\), since, according to (2.4), in this system \(u_1\ne 0\), which is impossible. Since \(2k-1\leq n\), for \(t\in [a_s,b_s]\) \(x(t)\) is a solution of the equation

\[ \frac{d^k}{dt^k}\left(x^{(n)}+u_n x^{(n-1)}+\cdots+u_{k+1}x^{(k)}\right) +u_k x^{(2k-1)}+\cdots+u_1 x^{(k)}=0. \]

Thus, the function \(x^{(k)}(t)\not\equiv 0\) satisfies an equation of type (2.5). According to the lemma,

\[ \left[\frac{\sigma_s}{n-1}\right]h_*\leq (b_s-a_s), \]

where \(\sigma_s\) is the number of zeros of \(x^{(k)}(t)\) on the interval \([a_s,b_s]\). Therefore

\[ \sum_s \left[\frac{\sigma_s}{n-1}\right]h_*\leq \sum_s (b_s-a_s)\leq m(T)h_*, \]

i.e.

\[ \sum_s \left[\frac{\sigma_s}{n-1}\right]\leq m(T). \tag{2.19} \]

Let \(s\in S_1\) if \(\sigma_s<n-1\), and \(s\in S_2\) if \(\sigma_s\geq n-1\). Then

\[ \sum_s \sigma_s = \sum_{s\in S_1}\sigma_s+\sum_{s\in S_2}\sigma_s \leq (n-1)\,[f_k(n)m(T)+1]+2(n-1)\times \]

\[ \times \sum_{s\in S_2}\left[\frac{\sigma_s}{n-1}\right] \leq (n-1)[f_k(n)+3]m(T). \]

Finally,

\[ r_{k+1}(T)=r_k(T)+\sum_s \sigma_s \leq f_{k+1}(n)m(T)=\{n f_k(n)+3(n-1)\}m(T). \]

The inductive argument carried out thus shows that the total number of zeros of the functions \(\psi_n(t),\ x(t),\ldots,\ x^{(\beta_n-1)}(t)\) \(\left(\beta_n=\left[\dfrac{n+3}{2}\right]\right)\) on any finite interval is finite and admits an estimate of the form (2.17), i.e. one not depending on the index \(p\). The theorem is thereby proved.

Let us examine in more detail the special case of equation (2.1) when \(m_i=M_i=c_i\) for \(2\leq i\leq n\). As follows from the proof of Theorem 3, in this case in the estimate (2.18) for the total number of zeros of \(x(t)\) and \(\psi_n(t)\), as \(h_*\) one may take any number having the property that, on the interval \([t_0,t_0+h_*]\), every solution of the equations

\[ x^{(n)}+c_n x^{(n-1)}+\cdots+c_2 x^{(1)}+u(t)x=0,\qquad m\leq u(t)\leq M \tag{2.20} \]

has no more than \((n-1)\) zeros. Suppose, for example, that all roots of the equations

\[ \lambda^n+c_n\lambda^{n-1}+\cdots+c_2\lambda+m=0,\qquad \lambda^n+c_n\lambda^{n-1}+\cdots+c_2\lambda+M=0 \tag{2.21} \]

are real—condition (A). Then (see, for example, the proof of the lemma in [1], § 17) any solution of equation (2.20) for \(u(t)=m\) or

\(u(t)=M\) on the interval \([0,+\infty)\) has no more than \((n-1)\) zeros. As follows from the results of [10], the same will also be true for an arbitrary admissible function \(u(t)\), \(m \leq u(t) \leq M\). Therefore, in inequality (2.18) the number \(h_*\) may be taken arbitrarily large, which leads to the following sufficient condition for the finiteness of the total number of switchings of the control \(u^{+\infty}(t)\).

Theorem 4. If condition (A) is satisfied, then among all controls \(u(t)\) that are optimal in the sense of the functional (2.2), there exists a control whose number of discontinuities does not exceed \((3n-2)\). Such a control is the function \(u^{+\infty}(t)\).

This result will be refined below.

Consider, instead of the functionals \(J(u)\) and \(J_p(u)\), respectively, the functionals

\[ J_T(u)=\max_{0\leq t\leq T}|x(t)|,\qquad J_{p,T}=\left(\int_0^T \rho(t)|x(t)|^p\,dt\right)^{1/p},\quad \rho>0,\quad \int_0^T \rho\,dt=1. \tag{2.22} \]

Let \(u_T^p(t)\) be a control optimal in the sense of the functional \(J_{p,T}(u)\). Analogously to the results of § 1, we obtain that the sequence of functions \(u_T^p(t)\) is minimizing for the functional \(J_T(u)\), and if by \(u_T^{+\infty}(t)\) we denote the limit of some subsequence \(u_T^{p_i}(t)\), then

\[ J_T\bigl(u_T^{+\infty}\bigr)=\inf_{u\in U} J_T(u),\qquad x(0)=x_0,\ldots,x^{(n-1)}(0)=x_0^{(n-1)}. \]

In turn, the sequence of functions \(u_T^{+\infty}(t)\) is minimizing for the functional \(J(u)\) in the sense that

\[ J_T\bigl(u_T^{+\infty}\bigr)\leq \inf_{u\in U} J(u)\quad\text{and}\quad \lim_{T\to+\infty} J_T\bigl(u_T^{+\infty}\bigr)= \]

\[ =\inf_{u\in U} J(u),\qquad x(0)=x_0,\ldots,x^{(n-1)}(0)=x_0^{(n-1)}. \]

Moreover, from any countable subsequence \(u_{T_s}^{+\infty}(t)\), \(T_s\to+\infty\), one can extract a subsequence \(u_{T_{s_i}}^{+\infty}(t)\) that converges weakly on every finite interval \([0,T]\) to \(u^{+\infty}(t)\), which is optimal in the sense of the functional (2.2); (we have in mind the convergence of the infinite tail of the sequence of functions \(u_{T_{s_i}}^{+\infty}\), for which \(T_{s_i}>T\)).

The control \(u_T^p(t)\) must satisfy the Pontryagin maximum principle (see (2.14), (2.15)) with the additional transversality condition \(\psi(T)=0\). Integrating the system \(\dot\psi=-A^*\psi-f(t)e_1\) by the method of variation of arbitrary constants, we obtain

\[ \psi(t)=\int_t^T \Psi(t,s)f(s)e_1\,ds, \]

where \(\Psi(t,s)\) is the matricant of the system \(\dot\psi(t)=-A^*(t)\psi(t)\), \(\Psi(s,s)=I_n\). Hence

\[ \psi_n(t)=(\psi(t),e_n)=\int_t^T f(s)(\Psi(t,s)e_1,e_n)\,ds; \]

but, as is known,

\[ \Psi^*(t,s)=Y^{-1}(t,s)=Y(s,t), \]

where \(Y(s,t)\) is the matricant of the system \(\dot y(s)=A(s)y(s)\), \(Y(t,t)=I_n\). We finally obtain

\[ \psi_n(t)=\int_t^T f(s)x_p(s,t)\,ds = \]

\[ = p\psi_0 \int_t^T \rho(s)|x(s)|^{p-1}\operatorname{sign} x(s)\,x_p(s,t)\,ds;\qquad \psi_0<0, \tag{2.23} \]

where \(x_p(s,t)\) is the solution of the equation

\[ \frac{\partial^n x(s,t)}{\partial s^n} + c_n \frac{\partial^{n-1}x(s,t)}{\partial s^{n-1}} + \cdots + u_T^p(s)x(s,t)=0, \tag{2.24} \]

\[ x(t,t)=0,\ldots,\left.\frac{\partial^{n-2}x(s,t)}{\partial s^{n-2}}\right|_{s=t}=0,\qquad \left.\frac{\partial^{n-1}x(s,t)}{\partial s^{n-1}}\right|_{s=t}=1. \]

From formulas (2.16), (2.23) it follows that the switching instants \(t_k^p(T)\) of the control \(u_T^p(t)\) are roots of one of the two equations:

\[ x(t_k)=0 \quad\text{or}\quad \int_{t_k}^T \rho(s)|x(s)|^{p-1}\operatorname{sign} x(s)\,x_p(s,t)\,ds=0. \tag{2.25} \]

Below we restrict ourselves to the consideration of the case in which condition (A) is satisfied. In this case, according to the foregoing and formula (2.24),

\[ x(s,t)>0 \quad \text{for all } s>t \text{ and for any admissible } u(s). \tag{2.26} \]

From (2.23), (2.26) it follows that all zeros of the function \(\psi_n(t)\), with the exception of the obvious zero at the point \(T\), lie to the left of the maximal zero of \(x(t)\). Further, as follows from the proof of Theorem 3, the equation for determining the function \(\psi_n(t)\) can be written in the form

\[ (-1)^n s_1(t)\frac{d}{dt}s_2(t)\cdots \frac{d}{dt}s_{n+1}(t)\psi_n = \psi_0 \rho p |x|^{p-1}\operatorname{sign} x; \]

\[ \psi_0<0,\qquad s_i>0. \tag{2.27} \]

The transversality conditions \(\psi(T)=0\) are equivalent to the conditions

\[ \psi_n(T)=\psi_n^{(1)}(T)=\cdots=\psi_n^{(n-1)}(T)=0. \tag{2.28} \]

Denote by \(r_x(T)\), \(r_{\psi_n}(T)\), respectively, the number of zeros of \(x(t)\) and \(\psi_n(t)\) inside the interval \([0,T]\). Then, by virtue of (2.28), the function \(\dfrac{d}{dt}(s_{n+1}\psi_n)\) will have at least \(r_{\psi_n}\) zeros inside \([0,T]\) and one zero at the point \(T\), etc.; finally, the same will be true for the function

\[ s_2(t)\frac{d}{dt}s_3(t)\cdots \frac{d}{dt}s_{n+1}\psi_n. \]

According to (2.27), this leads to the estimate

\[ r_{\psi_n}(T)\le r_x(T). \tag{2.29} \]

Using (2.16), we obtain: the number of switchings of the control \(u_T^p(t)\) does not exceed \(2r_x(T)\). Since always \(r_x(T)\le n-1\), more roughly, the number of switchings of \(u_T^p(t)\) does not exceed \(2(n-1)\). Moreover, according to (2.16), (2.23), \(u_T^p(t)\equiv M\) on the entire interval of sign-constancy of \(x(t)\) adjacent to the point \(t=T\).

It follows from the preceding that

Theorem 5. Among the controls that are optimal in the sense of the functional (2.2), there exists a control whose number of switchings does not exceed \(2(n-1)\).

Suppose now that a control \(u(t)\) minimizing the functional (2.2) exists for arbitrary initial conditions, and, in particular, for the following ones:

\[ x(0)=\ldots=x^{(n-2)}(0)=0,\qquad x^{(n-1)}(0)=1. \tag{2.30} \]

Then \(x(t)>0\) for all \(t>0\) and for any admissible \(u(t)\). Therefore, for any \(T\), \(\rho\), and \(p\), \(u_T^p(t)\equiv M,\ t\in[0,T]\), and if \(x_*(t)\) is a solution giving the minimum to the functional (2.2), then

\[ \begin{aligned} J_{p,T}(u_T^p) &=\left(\int_0^T \rho(t)|x_0(t)|^p\,dt\right)^{1/p} \leq \left(\int_0^T \rho(t)|x_*(t)|^p\,dt\right)^{1/p} \\ &\leq \max_{0\leq t\leq T}|x_*(t)| \leq \sup_{0\leq t<+\infty}|x_*(t)|; \end{aligned} \]

here \(x_0(t)\) is the solution of (2.20) with \(u(t)\equiv M\) and with initial conditions (2.30). Passing in this inequality to the limit first as \(p\to+\infty\), and then as \(T\to+\infty\), we obtain, taking into account the definition of \(x_*(t)\),

\[ \sup_{0\leq t<+\infty}|x_0(t)| = \sup_{0\leq t<+\infty}|x_*(t)|<+\infty. \]

It can be shown that the last inequality holds if and only if all roots of the equation

\[ \lambda^n+c_n\lambda^{n-1}+\ldots+c_2\lambda+M=0 \]

are nonpositive—condition (B). On the other hand, the usual arguments in such cases show that condition (B) is sufficient for the existence of trajectories that are optimal in the sense of the functional (2.2), with arbitrary initial data. Thus, the following theorem is valid.

Theorem 6. Let condition (A) be satisfied. Condition (B) is necessary and sufficient for the existence of trajectories, optimal in the sense of the functional (2.2), with arbitrary initial data.

It also follows from the preceding that, for the initial conditions (2.30) and the functionals \(J(u)\), \(J_T(u)\), \(J_{p,T}(u)\), the optimal control will be \(u(t)\equiv M\).

As for determining the control \(u_T^p(t)\), one may, without directly solving equations (2.25), search for the switching instants \(t_k^p(T)\), minimizing the functional \(J_{p,T}(u)\) by the descent method. Following the method of control variation proposed in [1], for computing the gradient of \(J_{p,T}(u)\) with respect to the variables \(t_k\), one can obtain the following formulas:

\[ \frac{\partial J_{p,T}(u)}{\partial t_k} = \left( \Delta u(t_k)x(t_k) \int_{t_k}^{T}\rho(t)|x(t)|^{p-1}x(t,t_k)\operatorname{sign}x(t)\,dt \right) \bigl(J_{p,T}(u)\bigr)^{1-p}, \]

\[ \Delta u(t_k)=u(t_k+0)-u(t_k-0). \]

(Equations (2.25) follow once again from these formulas.)

One may proceed differently: namely, passing to the limit as \(p\to+\infty\) in equations (2.25), obtain directly equations determining the switching instants of the control \(u_T^{+\infty}(t)\). Suppose, for example, that the initial conditions are

\[ x(0)=\ldots=x^{(n-3)}(0)=0;\qquad |x^{(n-2)}(0)|+|x^{(n-1)}(0)|\neq0. \tag{2.31} \]

(Obviously, for all admissible controls on the interval \((0,\delta]\), where \(\delta\) is sufficiently small, the corresponding \(x(t)\) is either always positive or always negative, depending on \(x^{(n-2)}(0)\), \(x^{(n-1)}(0)\). Let us assume, for definiteness, that \(x(t)>0\) for \(t\in(0,\delta]\) and for any admissible \(u(t)\). It is then clear that for any \(T\) and \(p\), \(r_{x_p}(T)\leqslant 1\), and the number of switchings of the control \(u_T^p(t)\) does not exceed two, \((x_p(t)\) being the solution corresponding to \(u_T^p(t))\). Denote by \(t_p\) and \(\tau_p\) the possible zeros of \(\psi_p(t)\) and \(x_p(t)\). The following cases may occur:

1) There exists a sequence \(p_i\to+\infty\) such that \(x_{p_i}(t)>0\) for \(t\in(0,T)\). Then \(u_T^{p_i}(t)\equiv M\), and \(u_T^{+\infty}(t)\equiv M\) for \(t\in[0,T]\).

2) For all sufficiently large \(p\) there exists \(\tau_p\in(0,T)\). Let \(p_i\) be a subsequence such that
\[ \lim_{p_i\to+\infty}\tau_{p_i}=\lim_{p\to+\infty}\tau_p=\tau_0 \]
and
\[ \lim_{p_i\to+\infty} t_{p_i}=t_0 \]
(if the \(t_{p_i}\) exist). Clearly, \(t_0\leqslant\tau_0\). Since \(\inf_p \tau_p>0\), we have \(\tau_0>0\).

a) \(\tau_0=T\). Then
\[ x_{+\infty}(t)=\lim_{p_i\to+\infty}x_{p_i}(t)>0 \quad \text{for } t\in(0,T), \qquad x_{+\infty}(T)=0. \]

We shall show that if the \(t_{p_i}\) exist for sufficiently large \(p_i\), then \(t_0=\tau_0=T\). Indeed, equation (2.25) may be written in the form

\[ \int_{t_{p_i}}^{\tau_{p_i}} \rho(s)|x_{p_i}(s)|^{p_i-1}x_{p_i}(s,t_{p_i})\,ds = \int_{\tau_{p_i}}^{T} \rho(s)|x_{p_i}(s)|^{p_i-1}x_{p_i}(s,t_{p_i})\,ds, \]

or

\[ \left( \int_{t_{p_i}}^{\tau_{p_i}} \rho(s)x_{p_i}(s,t_{p_i})|x_{p_i}(s)|^{p_i-1}\,ds \right)^{\frac{1}{p_i-1}} = \]

\[ = \left( \int_{\tau_{p_i}}^{T} \rho(s)x_{p_i}(s,t_{p_i})|x_{p_i}(s)|^{p_i-1}\,ds \right)^{\frac{1}{p_i-1}}. \tag{2.32} \]

Furthermore, for any \(\varepsilon_0<T\),

\[ \lim_{p_i\to+\infty} \left( \int_{\tau_{p_i}}^{T} \rho(s)x_{p_i}(s,t_{p_i})|x_{p_i}(s)|^{p_i-1}\,ds \right)^{\frac{1}{p_i-1}} \leqslant \]

\[ \leqslant \lim_{p_i\to+\infty} \left( \int_{T-\varepsilon_0}^{T} \rho(s)|x_{p_i}(s)|^{p_i-1}\,ds \right)^{\frac{1}{p_i-1}} = \max_{T-\varepsilon_0\leq t\leq T}|x_{+\infty}(t)|, \]

therefore the limit of the right-hand side of equality (2.32) is equal to zero. But if
\[ \tau_0-t_0=\varepsilon_1>0, \]
then for \(p_i>N\), where \(N\) is sufficiently large,

\[ \left( \int_{t_{p_i}}^{\tau_{p_i}} \rho(s)x_{p_i}(s,t_{p_i})|x_{p_i}(s)|^{p_i-1}\,ds \right)^{\frac{1}{p_i-1}} > \left( \int_{t_0+\frac{1}{3}\varepsilon_1}^{\tau_0-\frac{1}{3}\varepsilon_1} \rho(s)x_{p_i}(s,t_{p_i})|x_{p_i}(s)|^{p_i-1}\,ds \right)^{\frac{1}{p_i-1}}, \]

and (recall that \(x_{p_i}(t_{p_i},t_{p_i})=0\)),

\[ \inf_{\substack{t_0+\frac{\varepsilon_1}{3}\leq t\leq \tau_0-\frac{\varepsilon_1}{3}\\ p_i>N}} x_{p_i}(t,t_{p_i})\geqslant \alpha>0. \]

Therefore, on the other hand, for the left-hand side of equality (2.32)

\[ \lim_{p_i\to+\infty}\left(\int_{t_{p_i}}^{\tau_{p_i}} \rho(s)x_{p_i}(s,t_{p_i})|x_{p_i}(s)|^{p_i-1}\,ds\right)^{\frac{1}{p_i-1}} \geq \max_{t_0+\frac{\varepsilon_1}{3}\leq t\leq \tau_0-\frac{\varepsilon_1}{3}} |x_{+\infty}(t)|>0, \tag{2.33} \]

we have arrived at a contradiction. Thus \(\tau_0=t_0\).

It follows from the foregoing that either \(u_T^{+\infty}(t)\equiv M\), or \(u_T^{+\infty}(t)\equiv m\). We shall show more precisely that \(u_T^{+\infty}(t)\equiv M\). Indeed, otherwise, rewriting the equation for determining \(x_{+\infty}(t)\) in the form

\[ L_\delta(x)=x^{(n)}+c_n x^{(n-1)}+\cdots+c_2x^{(1)}+(m+\delta)x=\delta x, \]

\[ 0\leq \delta\leq M-m, \]

we obtain

Fig. 1

\[ x_{+\infty}(t)=x_\delta(t)+\delta\int_0^t K_\delta(t,s)x_{+\infty}(s)\,ds, \]

or

\[ x_\delta(t)=x_{+\infty}(t)-\delta\int_0^t K_\delta(t,s)x_{+\infty}(s)\,ds. \tag{2.34} \]

Here \(x_\delta(t)\), \(K_\delta(t,s)\) are solutions of the equation \(L_\delta(x)=0\), respectively with initial conditions (2.31) and the conditions

\[ K_\delta^{(l)}(s,s)= \left.\frac{\partial^l}{\partial t^l}K_\delta(t,s)\right|_{t=s}=0,\quad 1\leq l\leq n-2, \]

\[ \left.\frac{\partial^{n-1}}{\partial t^{n-1}}K_\delta(t,s)\right|_{t=s}=1. \]

It is obvious that \(K_\delta(t,s)>0\) for \(t>s\). Let \(\delta>0\). According to (2.34), \(x_\delta(t)<x(t)\) for \(t>0\), and for sufficiently small \(\delta\)

\[ 0<-\min_{0\leq t\leq T}x_\delta(t)<\max_{0\leq t\leq T}x_\delta(t)<\max_{0\leq t\leq T}x_{+\infty}(t); \]

hence it follows that the solution \(x_{+\infty}(t)\) is not optimal—a contradiction. Thus, \(u_T^{+\infty}(t)\equiv M\).

b) \(\tau_0<T\). If \(t_{p_i}\) do not exist, then the control \(u_T^{+\infty}(t)\) has the form

\[ u_T^{+\infty}(t)= \begin{cases} m, & \text{for } t\in[0,\tau_0],\\ M, & \text{for } t\in(\tau_0,T] \end{cases} \quad\text{(see Fig. 1).} \tag{2.35} \]

Let now \(t_{p_i}\) exist for sufficiently large \(p_i\). Similarly to item a), it is shown that \(t_0-\tau_0=-\Delta<0\). Since, for sufficiently large \(N\) and \(0<\varepsilon<\dfrac{\Delta}{2}\),

\[ \inf_{t_0+\varepsilon<t<T,\;p_i\geq N} x_{p_i}(t,t_{p_i})\geq \beta>0, \]

then

\[ \lim_{p_i\to+\infty}\left(\int_{\tau_{p_i}}^T \rho(s)x_{p_i}(s,t_{p_i})|x_{p_i}(s)|^{p_i-1}\,ds\right)^{\frac{1}{p_i-1}} = \max_{\tau_0\leq t\leq T}|x_{+\infty}(t)|, \]

and

\[ \lim_{p_i\to+\infty}\left(\int_{t_{p_i}}^{\tau_{p_i}} \rho(s)x_{p_i}(s,t_{p_i})|x_{p_i}(s)|^{p_i-1}\,ds\right)^{\frac1{p_i-1}} \ge \]

\[ \ge \lim_{p_i\to+\infty}\left(\int_{t_0+\varepsilon}^{\tau_{p_i}} \rho(s)|x_{p_i}(s)|^{p_i-1}\,ds\right)^{\frac1{p_i-1}} = \max_{t_0+\varepsilon\le t\le \tau_0}|x_{+\infty}(t)|, \]

i.e.

\[ \lim_{p_i\to+\infty}\left(\int_{t_{p_i}}^{\tau_{p_i}} \rho(s)x_{p_i}(s,t_{p_i})|x_{p_i}(s)|^{p_i-1}\,ds\right)^{\frac1{p_i-1}} = \max_{t_0+\varepsilon\le t\le \tau_0}|x_{+\infty}(t)|, \]

since the reverse inequality is obvious. Finally, the equation for determining the instant \(t_0\) takes the form

\[ \max_{t_0\le t\le \tau_0}|x_{+\infty}(t)| = \max_{\tau_0\le t\le T}|x_{+\infty}(t)|, \tag{2.36} \]

and the control \(u_T^{+\infty}(t)\) is determined by the equality

Fig. 2

\[ u_T^{+\infty}(t)= \begin{cases} M & \text{for } 0\le t\le t_0,\\ m & \text{for } t_0<t\le \tau_0,\\ M & \text{for } \tau_0<t\le T, \end{cases} \qquad x(\tau_0)=0 \quad \text{(see Fig. 2).} \]

Let us note that the control \(u_T^{+\infty}(t)\) will certainly be of this form if

\[ \max_{0\le t\le \tau_0}|x_{+\infty}(t)| > \max_{\tau_0\le t\le T}|x_{+\infty}(t)|. \]

This is proved in exactly the same way as in item a), where it was proved that \(u_T^{+\infty}(t)\equiv M\).

Remark. Everything set forth above is easily carried over to the case when the deviation from the zero solution is assigned a “weight” depending on time, i.e., to the case of the functional

\[ J(u)=\sup_{0\le t<+\infty}\sigma(t)|x(t)|,\qquad \sigma(t)>0 \text{ for } t\in[0,+\infty). \]

References

1. Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mishchenko E. F. The Mathematical Theory of Optimal Processes. Fizmatgiz, 1961.
2. Krylov A. N. PMM, vol. XXIV, issue 2, 1960.
3. Dubovitskii A. Ya., Milyutin A. A. DAN SSSR, No. 4, 1963.
4. Dubovitskii A. Ya., Milyutin A. A. Avtomatika i telemekhanika, vol. XXIV, No. 12, 1963.
5. Pittel B. G. Avtomatika i telemekhanika, vol. XXIV, No. 9, 1963; vol. XXIV, No. 11, 1963.
6. Hardy G. G., Littlewood J. E., Polya G. Inequalities. IL, 1948.
7. Pittel B. G. Vestnik LGU, No. 1, 1964.
8. Sansone J. Ordinary Differential Equations. IL, 1, 1953.
9. Mammana G. Mathematische Zeitschrift, 33B., 2H., 1931.
10. Kondrat’ev V. A. On the oscillation of solutions of the equation \(y^{(n)}+p(x)y=0\). Proceedings of the Moscow Mathematical Society, 10, 1961.

Received by the editors
March 12, 1965

Computing Center of the Leningrad Branch
of the V. A. Steklov Mathematical Institute
of the Academy of Sciences of the USSR