On the Problem of Optimal Control of a Mathematical Pendulum

Yu. M. Filimonov

The problem is considered of bringing a mathematical pendulum as rapidly as possible into the state of stable equilibrium under the action of an external torque bounded in absolute value. It is assumed that the greatest admissible value of the absolute value of the external torque exceeds the greatest value of the torque of the pendulum’s gravitational force.

The assumptions adopted lead to the time-optimal control problem \([1, 2]\) for a system of the following form:

\[ \frac{dx}{dt}=y,\quad \frac{dy}{dt}=-a\sin x+u, \tag{0.1} \]

where \(0<a<1\) is a constant; \(u\) are piecewise-smooth functions ensuring that the point of the phase plane reaches the position of stable equilibrium and satisfying the condition

\[ |u|\leq 1. \tag{0.2} \]

1. Following the terminology of the monograph \([3]\), we shall call the phase trajectory of system (0.1) corresponding to \(u=1\) a \(P\)-arc, and the phase trajectory corresponding to \(u=-1\) an \(N\)-arc. The totality of \(P\)-arcs of system (0.1) is represented by the family of curves

\[ \frac{1}{2}y^2=a\cos x+x+C. \tag{1.1} \]

The totality of \(N\)-arcs of the same system is represented by the family

\[ \frac{1}{2}y^2=a\cos x-x+C. \tag{1.2} \]

We shall adopt the following notation. The trajectory of system (0.1) passing through the points of the phase plane \(z_1, z_2\) and consisting of a specified sequence of \(P\)- and \(N\)-arcs will be denoted by \(z_1PN\ldots Nz_2\). In this notation the alternation of the letters \(P\) and \(N\) indicates the alternation of \(P\)- and \(N\)-arcs on the trajectory. If it is necessary to indicate the intermediate points through which the trajectory passes, then we shall intersperse the adopted notation with an indication of such points. We shall agree to denote the time lengths of trajectories by \(t(z_1PN\ldots Nz_2)\).

It will be shown below that if the motion of the mathematical pendulum is represented in the cylindrical phase space \(-\pi<x\leq \pi,\ -\infty<y<\infty\), then for system (0.1) there exists a set of points, forming

forming on the phase cylinder a helical line \((L)\) passing through the point \(z(0,\pi)\), which is defined by the equality

\[ t(zNPO)=t(zPNO),\quad z\in L. \]

Let \(\Gamma\) denote that part of the \(P\)-arc passing through the origin on which \(y<0\); let \(\Gamma^{-}\) denote the part of the \(N\)-arc passing through the origin and satisfying the condition \(y>0\). The arcs \(\Gamma\) and \(\Gamma^{-}\) together form a helical line \((C)\) on the phase cylinder. The lines \(L\) and \(C\) divide the phase cylinder into two helical strips, one of these strips containing the point \(z_{+}(\pi,\sqrt{\pi-2\alpha})\). The solution of the problem posed may be formulated as follows.

In a certain domain containing the origin, the optimal trajectories of the problem posed are the trajectories \(zP\Gamma^{-}O\), if the point \(z\) belongs to the strip containing \(z_{+}\), and the trajectory \(zN\Gamma O\) in the opposite case. If the point lies on the helical line \(L\), then both of the indicated trajectories are optimal (Fig. 1).

1. Passing to the proof of the main assertion, let us establish the validity of some auxiliary inequalities. Let \(z_1(x_1,y_1)\) and \(z(x_1,y_1+\varepsilon)\) be two points of the phase plane lying on the same vertical line. Here \(\varepsilon\) is a positive constant. Denote by \(z_1^{*}\), \(z_2^{*}\) the points of intersection of the \(N\)-arcs passing through \(z_1\) and \(z_2\), respectively, with the \(Ox\)-axis. The inequality

\[ t(z_1Nz_1^{*})<t(z_2Nz_2^{*}) \tag{2.1} \]

holds.

Fig. 1. Diagram of optimal control of a mathematical pendulum on the unrolled phase cylinder

We first prove inequality (2.1) for small values of \(\varepsilon\). Along the \(N\)-trajectory passing through the point \(z_1\), form the system of linear equations in variations

\[ \delta\dot{x}=\delta y,\quad \delta\dot{y}=-\alpha\cos x\,\delta x. \tag{2.2} \]

We shall seek a solution of system (2.2) satisfying the initial conditions

\[ \delta x(0)=0,\quad \delta y(0)=1, \tag{2.3} \]

in the form of series arranged in powers of the parameter \(\alpha\):

\[ \delta x=\sum_{i=0}^{\infty}\delta x_i\alpha^i,\quad \delta y=\sum_{i=0}^{\infty}\delta y_i\alpha^i. \tag{2.4} \]

Substituting (2.4) into (2.2), we find that the functions of time \(\delta x_i,\delta y_i\) \((i=0,1\ldots)\) must satisfy the following system:

\[ \delta\dot{x}_i=\delta y_i,\quad \delta\dot{y}_i=-\cos x\,\delta x_{i-1}\ (i=1,2\ldots),\quad \delta\dot{y}_0=0. \tag{2.5} \]

The arbitrary constants in the successive integration of system (2.5), in view of the initial conditions (2.3), may be chosen as follows:

\[ \delta x_0(0)=0,\quad \delta y_0(0)=1,\quad \delta x_i(0)=\delta y_i(0)=0. \]

Taking this into account, on the basis of system (2.5) we obtain the following estimates of the quantities \(\delta y_i\) \((i=1,2\ldots)\) for an arbitrary time interval \([0,\tau]\):

\[ |\delta y_i|\leq \frac{\tau^{2i}}{(2i)!}\quad (i=1,2\ldots). \tag{2.6} \]

Further, from (2.4) we have

\[ \delta y \geq 1-\sum_{i=1}^{\infty}\frac{\tau^{2i}}{(2i)!}\alpha^i >1-\frac{1}{2}(\exp \tau^2\alpha-1), \tag{2.7} \]

Consequently, the inequality

\[ \delta y>0 \tag{2.8} \]

will hold along any \(N\)-trajectory lying in the upper half-plane, whose time duration \((\tau)\) satisfies the inequality

\[ \tau^2\alpha\leq \ln 3. \tag{2.9} \]

In what follows we shall restrict ourselves to considering the region \((G)\) that contains the origin and is filled by segments of \(P\)- and \(N\)-trajectories of such time duration.

Let \(\hat{\delta}y\) denote the projection onto the axis \(Oy\) of the actual deviation of the trajectory \(z_2Nz_2^*\) from the trajectory \(z_1Nz_1^*\). Then \(\hat{\delta}y\) can be represented in the form [4]

\[ \hat{\delta}y=\varepsilon\delta y+o(\varepsilon). \tag{2.10} \]

It follows from this that, for sufficiently small values of \(\varepsilon\), by virtue of inequality (2.8), throughout the entire motion along the trajectory \(z_1Nz_1^*\) the inequality

\[ \hat{\delta}y>0. \tag{2.11} \]

will hold.

Geometrically, inequality (2.11) means that when the point moving along the trajectory \(z_1Nz_1^*\) reaches the axis \(Ox\), the point moving along the trajectory \(z_2Nz_2^*\) still remains in the upper half-plane. The validity of inequality (2.1) for finite values of \(\varepsilon\) can be verified on the basis of the Heine–Borel lemma on covering a closed set.

Let \(z_1, z_2\) be the same points as in inequality (2.1); the \(P\)-arc issuing from the point \(z_1\) intersects the trajectory \(z_2Nz_2^*\) at some point \(z_3\), since in the upper half-plane the angular coefficient of the tangent at any point of a \(P\)-arc is positive, and of an \(N\)-arc negative. Comparing the time lengths of the trajectories \(z_1Pz_3\) and \(z_2Nz_3\) with the aid of the first equation of system (0.1) leads to the inequality

\[ t(z_1Pz_3)>t(z_2Nz_3). \tag{2.12} \]

Taking inequalities (2.1) and (2.12) into account, we obtain

\[ t(z_1Nz_1^*)<t(z_1Pz_3Nz_2^*). \tag{2.13} \]

Since \(P\)- and \(N\)-arcs are symmetric with respect to the axis \(Ox\), and any \(P\)-arc is the mirror reflection of some \(N\)-arc with respect to the axis \(Oy\), the consequence of the obtained inequalities is the inequalities

\[ t(z_1Pz_1^*)<t(z_2Nz_2^*), \tag{2.14} \]

\[ t\left(z_1 P z_1^*\right) < t\left(z_1 N z_3 P z_2^*\right), \tag{2.15} \]

where \(z_1, z_1^*, z_2, z_2^*, z_3\) are points of the phase plane that are the mirror images, with respect to the \(Oy\) axis, of the corresponding points in inequalities (2.1) and (2.13). Analogous inequalities are also valid for the lower half-plane of the \(z\)-plane.

The existence of an optimal control “in the small” for any point of the phase plane of system (0.1) follows from [2]. Indeed, in the present case the vector \(q\) has the form

\[ q = \begin{pmatrix} 0\\ 1 \end{pmatrix}, \]

and the vector \(Q\cdot q\) has the form

\[ Q\cdot q = \begin{pmatrix} 0, & 1\\ -\alpha \cos x, & 0 \end{pmatrix} \begin{pmatrix} 0\\ 1 \end{pmatrix} = \begin{pmatrix} 1\\ 0 \end{pmatrix} \]

and, consequently, these vectors are noncollinear. The constraint on the velocity of the uncontrolled system is satisfied,

\[ y^2+\sin^2 x \leq x^2+y^2 . \]

Also satisfied is the requirement for the existence of a control satisfying condition (0.2) and bringing the point \(z=z_0\) to the equilibrium position. Such controls, for example, are controls that carry out the motion \(z_0 NPO\) or \(z_0 PNO\). In this case, as proved in the cited work, system (0.1) possesses an optimal piecewise-smooth control \(u_0(t)\) of the form

\[ u^0(t)=\operatorname{sign}\lambda(t), \]

where \(\lambda(t)\) can vanish only at isolated points \(t\in[0,T]\); \(T\) is the optimal transition time.

The class of admissible controls for our problem is narrowed still further by taking into account the necessary optimality condition (in the time-optimal sense) for systems of type (0.1) belonging to Bushaw [3].

We shall call a trajectory of system (0.1) canonical if the alternation of \(P\)- and \(N\)-arcs on it, as time increases, is subject to the following rule: in the upper half-plane a transition is possible only from a \(P\)-arc to an \(N\)-arc, and in the lower half-plane—from an \(N\)-arc to a \(P\)-arc. Bushaw’s condition consists in the fact that an optimal trajectory must be canonical.

Let us prove that the trajectories \(\Gamma\) and \(\Gamma^{-}\) are optimal. Let \(z_1\) be an arbitrary point on \(\Gamma^{-}\). Comparing the time length of the segment of the trajectory \(z_1 NO\) with other possible canonical trajectories, on the basis of inequality (2.13) we find

\[ t(z_1 NO)<t(z_1 Pz_2 Nz_2^* \ldots O). \tag{2.16} \]

The ellipsis in the right-hand side of inequality (2.16) denotes any finite alternation of \(P\)- and \(N\)-arcs. Inequality (2.16) means that the trajectory \(\Gamma^{-}\) is optimal. The optimality of the trajectory \(\Gamma\) is verified analogously.

We shall establish the existence of the line \(L\) defined above. The reasoning will be carried out for the upper half-plane; for the lower half-plane it is similar.

Consider, in the upper half-plane of the phase plane \(z\), the strip bounded by \(\Gamma^{-}\) and the \(N\)-trajectory entering the point \(O_1(0,2\pi)\). A pendulum with initial data from this strip can be brought to the stable

stable equilibrium along two different canonical trajectories of the following form: \(zPNO\) and \(zNPO\). Denote

\[ \Theta(z)=T_1(z)-T_2(z), \tag{2.17} \]

where \(T_1(z)=t(zPNO)\), \(T_2(z)=t(zNPO)\). By virtue of the continuous dependence of the solutions of system (0.1), for \(u=1\) and \(u=-1\), on the initial data, the function \(\Theta(z)\) is continuous inside the indicated strip.

Consider an arbitrary segment parallel to the axis \(Oy\), lying between the axis \(Ox\) and the \(N\)-trajectory entering the point \(O_1\). Using the notation of Fig. 2, we shall prove that on this segment there exists a unique point at which the function \(\Theta(z)\) vanishes. On the basis of inequality (2.15) and its analogue for the lower half-plane, we have:

\[ T_1(z_1)>T_1(z_0)=T_2(z_0)> \]
\[ >T_2(z_1), \quad z_0(\pi,0), \]

i.e. \(\Theta(z_1)>0\). Further, at points of the segment \(z\) close to the point \(z_2\), it is evident that the inequality \(T_1(z)<T_2(z)\) holds, i.e. at such points the function \(\Theta(z)\) will take negative values. Consequently, on the segment there is a point \(z^*\) at which \(\Theta(z^*)=0\).

Fig. 2.

On the basis of inequality (2.1) and the analogue of inequality (2.13) for the lower half-plane we conclude: the function \(T_2(z)\), as the point moves in the direction from \(z_1\) to \(z_2\), increases; on the basis of an inequality of type (2.12), the function \(T_1(z)\) decreases. Consequently, in this process the function \(\Theta(z)\) is monotone. This means that the point \(z^*\) on the given segment is unique. Analogous arguments can be carried out for any segment parallel to the axis \(Oy\) and lying between \(\Gamma^{-}\) and the \(N\)-trajectory entering the point \(O_1\) (and belonging to the domain \(G\)). The set of points \(z^*\) forms the line \(L\). The line \(L\) divides the strip enclosed between \(\Gamma^{-}\) and the \(N\)-trajectory entering the point \(O_1\) into two strips. As follows from the preceding arguments, in the strip containing the point \(z_{+}(\pi,\sqrt{\pi-2\alpha})\), the function \(\Theta(z)\) takes only negative values, while in the strip not containing this point it takes positive values.

Let \(z\) be an arbitrary point of the strip containing the point \(z_{+}\). We shall show that the optimal control for this point is the control that realizes the motion \(zPNO\). Let us compare the time duration of such a trajectory with the canonical trajectories bringing \(z\) into the position of stable equilibrium. By virtue of the property of the function \(\Theta(z)\), we have

\[ t(zPNO)<t(zNPO), \tag{2.18} \]

\[ t(zPNO)<t(zNz^*\ldots PO), \tag{2.19} \]

where \(z^*\) is the point of intersection of the trajectory with the axis \(Ox\). The ellipsis in the right-hand side of inequality (2.19) denotes any finite alternation of \(P\)- and \(N\)-arcs. On the basis of inequalities (2.18) and (2.19) we conclude that the trajectory \(zPNO\) is optimal. The optimality of the control method \(zNPO\) for points lying in the considered ...

plane. This fully justifies the formulated solution of the pendulum problem.

1. Thus, the optimal control law has been found. To construct the optimal system (0.1), it is necessary to determine the position of the line \(L\) on the phase plane (phase cylinder). We shall here set up the differential equation of the line \(L\).

Suppose that the point \(z_0 \in L\) has undergone some displacement along this line. Then, by virtue of the property of the line \(L\), the equality

\[ dT_1 = dT_2, \tag{3.1} \]

must hold, where \(T_1(z)\), \(T_2(z)\) are the functions defined above. Thus, in order to derive the required differential equation it is necessary to express the partial derivatives
\(\dfrac{\partial T_1}{\partial x_0}\), \(\dfrac{\partial T_1}{\partial y_0}\), \(\dfrac{\partial T_2}{\partial x_0}\), \(\dfrac{\partial T_2}{\partial y_0}\) as functions of \(z_0\). Denote by \(x_1=x_1(x_0,y_0,t)\), \(y_1=y_1(x_0,y_0,t)\) the equation of the \(P\)-arc issuing from the point \(z_0\); by \(z_1^*\) the point of intersection of this \(P\)-arc with the \(N\)-arc entering the point \(O_1\). Also denote by \(x_2=x_2(x_0,y_0,t)\), \(y_2=y_2(x_0,y_0,t)\) the equation of the \(N\)-arc issuing from the point \(z_0\); by \(z_2^*\) the point of intersection of this arc with \(\Gamma\). Taking (1.1) and (1.2) into account, we find

\[ x_1^*=\frac{C_1-C_0}{2},\qquad y_1^*=\sqrt{\,2\alpha\cos\frac{C_1-C_0}{2}+C_0+C_1\,}, \]

\[ x_2^*=\frac{C_2-C_3}{2},\qquad y_2^*=\sqrt{\,2\alpha\cos\frac{C_2-C_3}{2}+C_2+C_3\,}, \tag{3.2} \]

\[ C_0=\frac{1}{2}y_0^2-\alpha\cos x_0-x_0,\qquad C_1=4\pi-\alpha, \]

\[ C_2=\frac{1}{2}y_0^2-\alpha\cos x_0+x_0,\qquad C_3=-\alpha. \]

We shall dwell in detail only on the computation of \(\dfrac{\partial T}{\partial x_0}\), since the computation of the remaining partial derivatives is carried out analogously. Suppose that, under the displacement \(\Delta z_0(\Delta x_0,0)\), the value \(T_1(z_0)\) receives the increment \(\Delta T_1\), and the point \(z_1^*\) is displaced by the amount \(\Delta z_1(\Delta x_1,\Delta y_1)\).

Represent the change \(\Delta T_1\) in the form of the sum

\[ \Delta T_1=\Delta T_{11}+\Delta T_{12}, \tag{3.3} \]

where

\[ \Delta T_{11}=t(z_0+\Delta z\, P z_1^*+\Delta z_1)-t(z_0 P z_1^*), \]

\[ \Delta T_{12}=t(z_1^* N z_1^*+\Delta z_1). \]

Owing to the continuous differentiability of solutions with respect to the initial data [5], one can write (for the displacement \(\Delta z_0\) chosen above)

\[ \Delta x_1=\frac{\partial \tilde{x}_1}{\partial x_0}\Delta x_0+ \frac{\partial \tilde{x}}{\partial t}\Delta T_{11}, \]

\[ \Delta y_1=\frac{\partial \tilde{y}}{\partial x_0}\Delta x_0+ \frac{\partial \tilde{y}_1}{\partial t}\Delta T_{11}, \tag{3.4} \]

where the sign \(\sim\) means that the corresponding derivatives are computed at intermediate points. On the other hand, since the point \(z_1^*+\Delta z_1\) lies on the same arc as the point \(z_1\), by the mean-value theorem one can write

\[ (\alpha\sin \tilde x+1)\Delta x_1+\tilde y\,\Delta y_1=0 . \tag{3.5} \]

From equations (3.4) and (3.5) we obtain the equality

\[ \frac{\Delta T_{11}}{\Delta x_0} \left[ \frac{\partial \tilde x_1}{\partial t}(\alpha\sin \tilde x+1) + \frac{\partial \tilde y_1}{\partial t}\tilde y_1 \right] = \]

\[ = -\left[ \frac{\partial \tilde x_1}{\partial x_0}(\alpha\sin \tilde x+1) + \frac{\partial \tilde y_1}{\partial x_0}\tilde y_1 \right]. \tag{3.6} \]

Passing in equality (3.6) to the limit \(\Delta x_0\to 0\) and observing that in this case the second factor on the left-hand side has a nonzero limit, we find

\[ \lim_{\Delta x_0\to 0}\frac{\Delta T_{11}}{\Delta x_0} = -\frac{ (\alpha\sin x_1^*+1)\left(\dfrac{\partial x_1}{\partial x_0}\right) + y_1^*\left(\dfrac{\partial y_1}{\partial x_0}\right) }{ 2y_1^* }, \tag{3.7} \]

where \(\left(\dfrac{\partial x_1}{\partial x_0}\right)\), \(\left(\dfrac{\partial y_1}{\partial x_0}\right)\) are the results of computing the corresponding partial derivatives at \(t_1=t(z_0 Pz_1^*)\). Further, from (3.2) we have

\[ \Delta x_1 = -\frac{\alpha\sin \tilde x-1}{2}\,\Delta x_0 . \tag{3.8} \]

Equality (3.8), together with the first equation of system (0.1) for \(u=-1\), gives

\[ \frac{\Delta T_{12}}{\Delta x_0} = -\frac{\alpha\sin \tilde x_1-1}{2\tilde y}. \tag{3.9} \]

Passing to the limit \(\Delta x_0\to 0\), we obtain

\[ \lim_{\Delta x_0\to 0}\frac{\Delta T_{12}}{\Delta x_0} = -\frac{\alpha\sin x_1^*-1}{2y_1^*}. \tag{3.10} \]

Equality (3.3), taking into account (3.7) and (3.10), leads to the expression

\[ \frac{\partial T_1}{\partial x_0} = -\frac{ (\alpha\sin x_1^*+1)\left(\dfrac{\partial x_1}{\partial x_0}\right) + y_1^*\left(\dfrac{\partial y_1}{\partial x_0}\right) + \alpha\sin x_1^*-1 }{ 2y_1^* }. \tag{3.11} \]

Similarly we find

\[ \frac{\partial T_1}{\partial y_0} = -\frac{ y_1^* + (\alpha\sin x_1^*+1)\left(\dfrac{\partial x_1}{\partial y_0}\right) + y_1^*\left(\dfrac{\partial y_1}{\partial y_0}\right) }{ 2y_1^* }, \]

\[ \frac{\partial T_2}{\partial x_0} = \frac{ (\alpha\sin x_2^*-1)\left(\dfrac{\partial x_2}{\partial x_0}\right) + y_2^*\left(\dfrac{\partial y_2}{\partial x_0}\right) + \alpha\sin x_2^*+1 }{ 2y_2^* }, \tag{3.12} \]

\[ \frac{\partial T_2}{\partial y_0} = \frac{ y_1^*+(a\sin x_2^*-1)\left(\dfrac{\partial x_2}{\partial y_0}\right) +y_2^*\left(\dfrac{\partial y_2}{\partial y_0}\right) }{2y_2^*}, \]

where \(\left(\dfrac{\partial x_1}{\partial y_0}\right)\), \(\left(\dfrac{\partial y_1}{\partial y_0}\right)\) are the results of computing the corresponding partial derivatives at \(t=t_1\), and

\[ \left(\frac{\partial x_2}{\partial x_0}\right),\quad \left(\frac{\partial y_2}{\partial x_0}\right),\quad \left(\frac{\partial x_2}{\partial y_0}\right),\quad \left(\frac{\partial y_2}{\partial y_0}\right) \]

are at \(t_2=t(z_0,Nz_2^*)\).

To compute the indicated values of the partial derivatives

\[ \frac{\partial x_i}{\partial x_0},\quad \frac{\partial y_i}{\partial x_0},\quad \frac{\partial x_i}{\partial y_0},\quad \frac{\partial y_i}{\partial y_0}\quad (i=1,2) \]

we note that, pairwise, they satisfy the following systems of ordinary differential equations [5]:

\[ \frac{d\varphi_i}{dt}=\psi_i,\qquad \frac{d\psi_i}{dt}=-a\cos x_i\,\varphi_i \quad (i=1,2). \tag{3.13} \]

Here the functions

\[ \varphi_i(t)=\frac{\partial x_i}{\partial x_0},\qquad \psi_i(t)=\frac{\partial y_i}{\partial x_0} \quad (i=1,2) \]

satisfy system (3.13) under the conditions \(\varphi_i(0)=1\), \(\psi_i(0)=0\), while the functions

\[ \varphi_i(t)=\frac{\partial x_i}{\partial y_0},\qquad \psi_i(t)=\frac{\partial y_i}{\partial y_0} \quad (i=1,2) \]

satisfy system (3.13) and the condition \(\varphi_i(0)=0\), \(\psi_i(0)=1\). Since \(y\) varies monotonically with time along the \(P\)- and \(N\)-arcs, then, taking \(y\) as the new independent variable and taking into account system (0.1) both for \(u=1\) and for \(u=-1\), systems (3.13) can be transformed to the form

\[ \frac{d\varphi_i}{dt} = \frac{-1}{a\sin x_i+(-1)^i}\,\psi_i, \]

\[ \frac{d\psi_i}{dt} = \frac{a\cos x_i}{a\sin x_i+(-1)^i}\,\varphi_i, \tag{3.14} \]

\[ \frac{dx_i}{dy} = \frac{-y}{a\sin x_i+(-1)^i} \quad (i=1,2). \]

Solutions of system (3.14) satisfying the conditions

\[ x_i(y_0)=x_0,\quad \varphi_i(y_0)=1,\quad \psi_i(y_0)=0 \quad (i=1,2), \]

give

\[ \left(\frac{\partial x_i}{\partial x_0}\right) = \varphi_i(y_i^*),\qquad \left(\frac{\partial y_i}{\partial x_0}\right) = \psi_i(y_i^*) \quad (i=1,2). \tag{3.15} \]

Having found the solutions of system (3.14) determined by the conditions

\[ \tilde{x}_i(y_0)=x_0,\quad \tilde{\varphi}_i(y_0)=0,\quad \tilde{\psi}_i(y_0)=1, \]

we shall have

\[ \left(\frac{\partial x_i}{\partial y_0}\right)=\tilde{\varphi}_i(y_i^*), \qquad \left(\frac{\partial y_i}{\partial y_0}\right)=\tilde{\psi}_i(y_i^*) \quad (i=1,2). \tag{3.16} \]

Thus, the equalities (3.2), (3.11), (3.12) and the indicated procedure for computing the quantities (3.15), (3.16) give the desired expression of the partial derivatives \(\dfrac{\partial T_i}{\partial x_0}\), \(\dfrac{\partial T_i}{\partial y_0}\) \((i=1,2)\) as functions of \(z_0\).

Let us write condition (3.1) in expanded form:

\[ \left(\frac{\partial T_1}{\partial y_0}-\frac{\partial T_2}{\partial y_0}\right)dy_0 = \left(\frac{\partial T_2}{\partial x_0}-\frac{\partial T_1}{\partial x_0}\right)dx_0 . \tag{3.17} \]

On the basis of inequality (2.1) and the analogue of inequality (2.13) for the lower half-plane, as well as an inequality of type (2.12), we conclude that the first factor on the left-hand side of (3.17) does not vanish (it is always negative). Therefore the differential equation of the line \(L\) can be written in the form

\[ \frac{dy_0}{dx_0} = \left(\frac{\partial T_2}{\partial x_0}-\frac{\partial T_1}{\partial x_0}\right) : \left(\frac{\partial T_1}{\partial y_0}-\frac{\partial T_2}{\partial y_0}\right). \tag{3.18} \]

To construct the line \(L\), the differential equation (3.18) should be integrated (by some numerical method) under the condition \(x_0=\pi\), \(y_0=0\).

References

1. Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mishchenko E. F. Mathematical Theory of Optimal Processes. Fizmatgiz, Moscow, 1961.
2. Krasovskii N. N. PMM, 23, no. 2, 222, 1959.
3. Tsyan Syue-sen. Technical Cybernetics. IL, 1956.
4. Nemytskii V. V., Stepanov V. V. Qualitative Theory of Differential Equations, 2nd ed. Gostekhizdat, Moscow—Leningrad, 1949.
5. Stepanov V. V. Course of Differential Equations, 5th ed. Gostekhizdat, Moscow—Leningrad, 1950.

Received by the editors
March 23, 1965

Nizhny Tagil State
Pedagogical Institute