UDC 517.919

THE JUMP CONDITION IN A PROBLEM OF OPTIMAL CONTROL

HOANG HUU DUONG

1. In the present paper the following problem is considered [1]. Let the hypersurface \(g(x)=0\) divide the \(n\)-dimensional space \(X\) into two parts \(X_1, X_2\). The motion of a point is determined by the systems:

\[ \frac{dx}{dt}=f_1(x,u)\quad \text{in } X_1, \]

\[ \frac{dx}{dt}=f_2(x,u)\quad \text{in } X_2. \]

The boundary conditions \(x(0)=x_0\in X_1,\ x(T)=x_1\in X_2\) are prescribed. It is required to choose a control \(u(t),\ 0\le t\le T\), in such a way that the corresponding trajectory \(x(t)\) minimizes the integral \(\int_0^T f^0(x(t),u(t))\,dt\). It was known that, upon crossing the hypersurface \(g(x)=0\), the so-called jump condition is satisfied; however, it was proved [2] only in the case when the trajectory \(x(t)\) does not touch the hypersurface \(g(x)=0\) from either side, i.e., when

\[ \begin{aligned} (\operatorname{grad} g(x(\tau)),\, f_1(x(\tau),u(\tau-0))) &\ne 0,\\ (\operatorname{grad} g(x(\tau)),\, f_2(x(\tau),u(\tau+0))) &\ne 0, \end{aligned} \tag{1} \]

where \(x(\tau)\) is the junction point of the optimal trajectory.

In the case when at least one of these scalar products vanishes, the jump condition had not been proved, and the existing method of proof did not carry over directly to this case.

In the present paper it is asserted that, under a certain assumption, the jump condition is satisfied independently of whether the inequalities (1) hold.

1. Consider case I:

\[ (\operatorname{grad} g(x(\tau)),\, f_1(x(\tau),u(\tau-0)))=0, \]

\[ (\operatorname{grad} g(x(\tau)),\, f_2(x(\tau),u(\tau+0)))\ne 0. \]

Introduce one more coordinate \(x^0\):

\[ \frac{dx^0}{dt}=f^0(x,u). \]

Consider the systems in the \(n+1\)-dimensional phase space \(\hat X(x^0,x)\):

\[ \frac{d\hat x}{dt}=\hat f_1(x,u)\quad \text{in } \hat X_1, \tag{2.1} \]

\[ \frac{d\hat{x}}{dt}=\hat{f}_2(x,u)\quad \text{in } \hat{X}_2, \tag{2.2} \]

where \(\hat{x}=(x^0,x)\), \(\hat{f}_i=(f^0,f_i)\). \(\hat{X}_1, \hat{X}_2\) are the two parts of the space \(\hat{X}\), separated by the hypersurface \(g(\hat{x})=g(x)=0\). Suppose that \(\hat{f}_1,\hat{f}_2\) are continuous and continuously differentiable, and that \(u(t)\), \(g(\hat{x})\) satisfy the conditions mentioned in [1]. Then we can formulate the optimal problem posed above in the following equivalent form.

In the space \(\hat{X}\) there are given a point \(\hat{x}_0=(0,x_0)\in \hat{X}_1\) and a straight line \(\Pi\), parallel to the \(x^0\)-axis and passing through the point \(\hat{x}_1=(0,x_1)\in \hat{X}_2\). It is required to choose an admissible control \(u(t)\) such that the phase point from the initial position \(\hat{x}_0\) reaches the line \(\Pi\), and the coordinate \(x^0\) of the end of the trajectory attains a minimum.

Suppose that the following conditions are satisfied.

1). Let \(\varepsilon\) be an arbitrary small number. Suppose that there exists a point \(x_2\), belonging to the boundary \(g(\hat{x})=0\), such that \(\|\hat{x}(\tau)-x_2\|=o(\varepsilon)\), and that there exists a control \(\bar u(t)\), \(\tau\leq t\leq t_1\), which transfers the phase point from the junction point \(\hat{x}(\tau)\) to the point \(x_2\) along a regular trajectory of system (2.1). Suppose that \(t_1-\tau\to 0\) as \(\varepsilon\to 0\).

2). There exists a control \(\bar{\bar u}(t)\) such that: a) the corresponding trajectory \(\bar{\bar x}(t)\) of system (2.2), satisfying the condition \(\bar{\bar x}(t_1)=x_2\), passes at time \(t_2\) through the point \(\bar{\bar x}(t_2)\in \Pi\) and has the unique common point \(x_2\) with the boundary \(g(\hat{x})=0\); b) \(\|\bar{\bar u}(t)-u(t)\|=o(\varepsilon)\), \(t_1\leq t\leq t_2\); c) \(|t_2-T|=o(\varepsilon)\). Then, by virtue of the continuous dependence of the solution on the parameters, we obtain

\[ \|\bar{\bar x}(t_2)-\hat{x}(T)\|=o(\varepsilon). \tag{2.3} \]

2.1. Consider the system

\[ \frac{d\xi}{d\theta}=-\hat{f}_1(\xi,v), \tag{2.4} \]

\(0\leq \theta\leq \tau+\varepsilon\delta\vartheta\), where \(\delta\vartheta\) is an arbitrary real number; \(\varepsilon>0\) is a sufficiently small number. The set of vectors \(\delta_1=A^1_{\tau,0}(\xi_0)+\Delta\xi_a^1\), for all possible symbols \(a\) and \(\xi_0\), is a convex cone \(K^1\) with vertex \(\hat{x}_0\). Here the vector \(\xi_0\) issues from \(\hat{x}(\tau)\) and is either equal to zero or directed into \(\hat{X}_1\) and does not touch the boundary \(g(\hat{x})=0\) at \(\hat{x}(\tau)\). \(A^1_{\tau,0}\) is the transport operator, corresponding to (2.4), along the trajectory and the control \(\xi(t)=x(\tau-t)\), \(v(t)=u(\tau-t)\); \(\Delta\xi_a\) is defined in [1].

We consider the system

\[ \frac{dy}{dt}=\hat{f}_1(y,v),\qquad R(y,v,\varepsilon\delta\mu)=0 \]

(where \(R(y,v,\varepsilon\delta\mu)\) is defined in [1]), corresponding to the segment of the regular trajectory \(\bar{x}(t)\), \(\tau\leq t\leq t_1\). Now consider the set of vectors \(\delta_2=P_{t_1,\tau}(\delta\mu)(\delta\hat{x}(\tau))\), where the symbol \(a\) (and also, let the symbol \(b\)) is defined in [1].

Construct a cone \(K^3\) with vertex \(\bar{\bar x}(t_2)\), corresponding to \(\bar{\bar x}(t)\), \(\bar{\bar u}(t)\) and filled by the set of vectors \(\delta_3=A^2_{t_2,t_1}(\bar{\delta}_2)+\Delta\hat{x}_a\), where \(A^2_{t_2,t_1}\), \(\Delta\hat{x}_a\) correspond to system (2.2).

With respect to the segment of the regular trajectory \(\bar{x}(t)\), we have a segment of the variation \(\bar{x}^{*}(t)\), \(\tau \leq t \leq t_1\), depending on the symbol \(b\). Note that \(\bar{x}^{*}(t_1)\) belongs to the boundary \(g(\hat{x})=0\). Next we construct the variation \(\bar{\bar{x}}^{*}(t)\) of \(\bar{\bar{x}}(t)\), corresponding to the symbol \(a\) with the initial condition

\[ \bar{\bar{x}}^{*}(t_1)=\bar{\bar{x}}(t_1)+\bigl(\bar{x}^{*}(t_1)-\bar{x}(t_1)\bigr)=\bar{x}^{*}(t_1). \]

Note that in the case under consideration \(\bar{\bar{x}}^{*}(t)\) lies entirely in \(\hat{X}_2\) for \(t_1<t\leq t_2\). Since \(x^{*}(t_1)-\bar{x}(t_1)=\varepsilon \delta_2+o(\varepsilon)\), we obtain

\[ \bar{\bar{x}}^{*}(t_2+\varepsilon\delta t)-\bar{\bar{x}}(t_2) =\varepsilon\bigl[A^2_{t_2,t_1}(\delta_2)+\Delta\hat{x}_a\bigr]+o(\varepsilon). \tag{2.5} \]

On the other hand, consider the segment of the trajectory \(\xi(t)\), \(0\leq t\leq \tau+\varepsilon\delta\vartheta\), of system (2.4). With respect to the variation \(\xi^{*}(t)\) we have

\[ \xi^{*}(\tau+\varepsilon\delta\vartheta)-\xi(\tau) =\varepsilon\bigl[A^1_{\tau,0}(\xi_0)+\Delta\bar{\xi}_a\bigr]+o(\varepsilon). \tag{2.6} \]

2.2. We now construct a convex cone \(K\subset K^1\times K^3\) corresponding to \(\xi_0=\delta\hat{x}(\tau)\). By the method of [1] it can be proved that \(K\) is separated from the ray \(\hat{x}_0\times L\), where \(L\) is the ray issuing from \(\bar{\bar{x}}(t_2)\) and going in the direction of the negative half-axis \(x^0\). Indeed, otherwise, by virtue of (2.5), (2.6), there would exist a continuous trajectory issuing from \(\hat{x}_0\) and satisfying (2.1), (2.2), respectively in \(\hat{X}_1,\hat{X}_2\), such that it passes through a point \(x^{*}\in L\), at a distance \(\varepsilon d+o(\varepsilon)\) from \(\bar{\bar{x}}(t_2)\), where \(d\) does not depend on \(\varepsilon\). On the other hand, it follows from (2.3) that the coordinate \(x^0\) of \(x^{*}\) is less than \(x^0(T)\) for sufficiently small \(\varepsilon\). But the latter assertion contradicts the optimality of the trajectory \(\hat{x}(t)\).

Thus, there exists a nonzero vector \((\bar{\chi}_1,\bar{\chi}_2)\) in \(\hat{X}_1\times\hat{X}_2\) such that

\[ (\bar{\chi}_1,\delta_1)+(\bar{\chi}_3,\delta_3)\leq 0. \tag{2.7} \]

2.3. Consider the system

\[ \frac{d\xi}{dt}=\frac{\partial\hat{f}_1\bigl(\hat{x}(t),u(t)\bigr)}{dx}\,\xi,\quad 0\leq t\leq \tau. \]

Let \(\zeta(t)\) be a solution satisfying the initial condition \(\zeta(\tau)=\chi_1\). Then, with the empty symbol \(a\) and \(\xi_0=-N\), where \(N\) is an arbitrary vector not tangent to the boundary \(g(\hat{x})=0\) at \(\hat{x}(\tau)\) and belonging to \(\hat{X}_2\), we obtain

\[ (\bar{\chi}_1,\delta_1)=(\tilde{\Psi}^{-}(\tau),N), \tag{2.8} \]

where \(\tilde{\Psi}(t)=-\zeta(\tau-t)\) is a solution of the system

\[ \frac{d\Psi^{-}}{dt}=-\frac{\partial\hat{f}_1\bigl(\hat{x}(t),u(t)\bigr)}{\partial x}\Psi^{-}. \tag{2.9} \]

Now consider the system

\[ \frac{d\Psi}{dt}=-\frac{\partial\hat{f}_2\bigl(\bar{\bar{x}}(t),\bar{\bar{u}}(t)\bigr)}{\partial x}\Psi. \]

Let \(\overline{\Psi}(t)\) be a solution satisfying the condition \(\overline{\Psi}(t_2)=\overline{\chi}_3\). Then with empty \(a\) we obtain

\[ (\overline{\chi}_3,\delta_3)=(\overline{\Psi}(t_1),\delta_2). \tag{2.10} \]

Consider the system

\[ \frac{d\Psi}{dt} = -\left( \frac{\partial \hat f_1(\overline{x}(t),\overline{u}(t))}{\partial x} + \overline{\Lambda}(t) \frac{\partial p(\overline{x}(t),\overline{u}(t))}{\partial x} \right)\Psi, \tag{2.11} \]

where \(\hat{\Lambda}(t)\), \(p(x,u)\) have the same meaning as in [1], for \(\tau\le t\le t_1\), \(\hat f=\hat f_1\). Let \(\Psi(t)\) be a solution of system (2.11) satisfying the condition

\[ \Psi(t_1)=\overline{\Psi}(t_1). \tag{2.12} \]

With empty symbol \(a\) and with \(b=(\xi_1=\hat{x}(\tau),a_1(x),N,1)\), it follows from (2.10) that

\[ (\overline{\chi}_3,\delta_3) = -(\Psi(\tau),N) + \overline{\lambda}(\tau) \left( \frac{\partial g(\overline{x}(\tau))}{\partial x},N \right) + \]

\[ \quad + \int_{\tau}^{t_1} \frac{d\overline{\lambda}}{dt} a_1(\overline{x}(t)) \left( \frac{\partial g(\overline{x}(t))}{\partial x},N \right)\,dt, \tag{2.13} \]

where \(\overline{\lambda}(t)=-(\Psi(t),\overline{\Lambda}(t))\). From (2.7), (2.8), (2.13), with sufficiently small \(O_\varepsilon\), we obtain

\[ (\overline{\Psi}(\tau)-\Psi(\tau)+\overline{\lambda}(\tau)\operatorname{grad} g(\hat{x}(\tau)),N)\le 0. \tag{2.14} \]

2.4. The set \(\delta_3=\Delta\hat{x}_a\) fills a convex cone \(k_3\). Since \(K\supset x_1\times k_3\), we can prove, as in Theorem 8 of [1], that:

1) \[ H(\overline{\Psi}(t),\overline{x}(t),\overline{u}(t)) = \max_u H(\overline{\Psi}(t),\overline{x}(t),u) = M(\overline{\Psi}(t),\overline{x}(t)); \]

2) \(\overline{\Psi}^{0}(t)=\mathrm{const}\le 0\), \(M(\overline{\Psi}(t),\overline{x}(t))=0\) for \(t_1\le t\le t_2\). As \(\varepsilon\to0\), \(t_1\to\tau\), \(t_2\to T\), and we have uniformly \(\overline{x}(t)\to x(t)\), \(\overline{u}(t)\to u(t)\). Further, \(\overline{\chi}\to\chi=(\chi_1,\chi_3)\ne0\), since we can choose \(\|\overline{\chi}\|=1\). Then \(\overline{\Psi}(t)\) tends to the solution \(\Psi^{+}(t)\) of the system

\[ \frac{d\Psi^{+}}{dt} = - \frac{\partial \hat f_2(x(t),u(t))}{\partial x}\Psi^{+}, \qquad \tau\le t\le T, \tag{2.15} \]

with \(\Psi^{+}(T)=\chi_3\); consequently, \(\Psi^{+}(t)\) satisfies conditions 1), 2) of the maximum principle [1]. As \(\varepsilon\to0\) we have \(\overline{\Psi}(t_1)\to\Psi^{+}(\tau)\), \(\Psi(t_1)\to\Psi(\tau)\). Then from (2.12) we obtain \(\Psi(\tau)=\Psi^{+}(\tau)\). On the other hand, as \(\varepsilon\to0\), \(\Psi(t)\) tends to the solution \(\Psi^{-}(t)\) of system (2.9), satisfying the initial condition \(\Psi^{-}(0)=-\chi_1\). Therefore, from (2.14) we obtain

\[ (\Psi^{-}(\tau)-\Psi^{+}(\tau)+\lambda(\tau)\operatorname{grad} g(\hat{x}(\tau)),N)\le0, \]

where \(\lambda(t)=\lim_{\varepsilon\to0}\overline{\lambda}(t)\). In view of the arbitrariness of the vector \(N\), it follows that

\[ \Psi^{+}(\tau)=\Psi^{-}(\tau)+\mu\operatorname{grad} g(\hat{x}(\tau)), \tag{2.16} \]

where \(\mu=\mathrm{const}\). We note that \(K\supset \overline{x}(\hat{t}_3)\times k_1\), where \(k_1\) is a convex cone filled by the set of vectors \(\delta_1\) for \(\xi_0=0\). Consequently, \(\Psi^{-}(t)\) satisfies conditions 1), 2) of the maximum principle [1].

2.5. Thus, if \(\chi_1\ne 0\), \(\chi_3\ne 0\), then \(\Psi^{-}(t)\), \(\Psi^{+}(t)\) satisfy the maximum principle. In other words, (2.16) represents the jump condition in the first case. If \(\chi_1=0\), then \(\chi_3\ne 0\). In this case \(\Psi^{-}(t)=0\), and \(\Psi^{+}(t)\) satisfies the maximum principle. Consequently, from (2.16) we obtain the jump condition in the second case

\[ \Psi^{+}(\tau)=\mu\,\operatorname{grad} g(\hat{x}(\tau)). \tag{2.17} \]

If \(\chi_3=0\), then \(\chi_1\ne 0\), and we have the jump condition in the third case

\[ \Psi^{-}(\tau)=-\mu\,\operatorname{grad} g(\hat{x}(\tau)). \tag{2.18} \]

1. Now consider case II:

\[ (\operatorname{grad} g(x(\tau)),\, f_1(x(\tau), u(\tau-0)))\ne 0, \]

\[ (\operatorname{grad} g(x(\tau)),\, f_2(x(\tau), u(\tau+0)))=0. \]

We set \(\vartheta=T-t\). Then systems (2.1), (2.2) become

\[ \frac{d\xi}{d\vartheta}=-\hat f_1(\xi,v),\qquad \frac{d\xi}{d\vartheta}=-\hat f_2(\xi,v), \]

and \(J\) becomes \(\displaystyle \int_0^T f^0(\xi,v)\,d\vartheta\). Thus we return to case I.

1. Consider case III:

\[ (\operatorname{grad} g(x(\tau)),\, f_1(x(\tau), u(\tau-0)))=0, \]

\[ (\operatorname{grad} g(x(\tau)),\, f_2(x(\tau), u(\tau+0)))=0. \]

Suppose that:

1). Let \(\hat O x(\tau)\) be a sufficiently small \(\varepsilon\)-neighborhood of a point belonging to the boundary \(g(\hat x)=0\). Then there exists a control \(u_1(t)\), \(\tau\leq t\leq t_1\), which transfers the phase point from \(\hat x(\tau)\) to \(x_2\in \hat O x(\tau)\) along a regular trajectory \(x_1(t)\) of system (2.1). Suppose that \(t_1-\tau\to 0\) as \(\varepsilon\to 0\). Let \(P'_{t_1,\tau}\) be the transport operator corresponding to this segment of the trajectory. We denote

\[ \delta'_2=P'_{t_1,\tau}(\delta\mu)\,\delta x_1(\tau), \tag{4.1} \]

where

\[ \delta x_1(\tau)=-\delta\mu\sum_{\alpha=1}^{s} a_{1\alpha}(x(\tau))N_{1\alpha} \]

for \(b_1=\{\zeta_{1i}, a_{1i}(x_1), N_{1i}, \delta\mu\}\) \((i=1,\ldots,s)\),

\(N_{1i}\) are vectors lying in \(X_2\) and not tangent to the boundary \(g(\hat x)=0\). The index 1 means that the symbol \(b\) corresponds to \(x_1(t)\). The symbol \(a\) is empty.

2). There exists a control \(u_2(t)\), \(t_1\leq t\leq t'_1\), which transfers the phase point from \(x_2\) to \(\hat x(\tau)\) along a regular trajectory \(x_2(t)\) of system (2.2). Suppose that \(t'_1-t_1\to 0\) as \(\varepsilon\to 0\). Setting \(\vartheta=t'_1-t\), from (2.2) we obtain

\[ \frac{d\xi}{d\vartheta}=-\hat f_2(\xi,v),\quad 0\leq \vartheta\leq t'_1-t_1=\vartheta_1. \]

Consider the trajectory and the control

\[ \xi(\vartheta)=x_2(t'_1-\vartheta),\quad v(\vartheta)=u_2(t'_1-\vartheta). \]

Let \(P_{\vartheta_1,0}^{2}\) be the transfer operator corresponding to the regular trajectory \(\xi(\vartheta)\). We denote

\[ \delta_2^2=P_{\vartheta_1,0}^{2}(\delta\mu)\,\delta\xi(0), \tag{4.2} \]

where

\[ \delta\xi(0)=-\delta\mu\sum_{\alpha=1}^{s} a_{2\alpha}(\xi(0))N_{2\alpha} \]

for \(b_2=\{\zeta_{2i}, a_{2i}(\xi), N_{2i}, \delta\mu\}\) \((i=1,\ldots,s)\);

\(N_{2i}\) are vectors lying in \(\hat X_1\) and not touching the boundary \(g(\hat x)=0\). The index 2 means that the symbol \(b\) corresponds to \(\xi(t)\). The symbol \(a\) is empty. We choose \(0\le a_{2i}(\xi)\le K\), where \(K\) is a fixed constant. At the instant \(t_1\) we have

\[ x_1^*(t_1)-x_1(t_1)=\varepsilon\delta_2' + o(\varepsilon), \tag{4.3} \]

where \(x_1^*(t)\) is a variation of the trajectory \(x_1(t)\). Note that \(x_1^*(t_1)\) lies on the boundary \(g(\hat x)=0\). If we take all possible symbols \(b_1\) under the conditions \(\|N_{1i}\|\le C\), where \(C\) is some positive constant, then the end of the vector \(x_1^*(t)-x_1(t_1)\) generates the set \(\Gamma_1\). Similarly, for the segment \(\xi(\vartheta)\), \(0\le \vartheta\le \vartheta_1\), at the instant \(\vartheta_1\) we obtain

\[ \xi^*(\vartheta_1)-\xi_1(\vartheta_1)=\varepsilon\delta_2^2+o(\varepsilon), \]

where \(\xi^*\) is a variation of \(\xi_1\). Note that \(\xi^*(\vartheta_1)\) lies on the boundary \(g(\hat x)=0\). Returning to the variable \(x\) and observing that \(x_2(t_1)=x_1(t_1')\), we obtain

\[ x_2^*(t_1)-x_1(t_1)=\varepsilon\delta_2^2+o(\varepsilon), \tag{4.4} \]

where \(x_2^*(t)=\xi^*(t_1'-\vartheta)\) is a variation of \(x_1(t)\). If we take all possible symbols \(b_2\), then the end of the vector \(x_2^*(t_1)-x_1(t_1)\) generates the set \(\Gamma_2\).

The symbol \(b_2\) of the trajectory \(\xi(\vartheta)\) is called the corresponding symbol \(b_1\) of the trajectory \(x_1(t)\) if the ends of the corresponding vectors \(\xi^*(\vartheta_1)-\xi(\vartheta_1)\) and \(x_1^*(t_1)-x_1(t_1)\) coincide.

3). Suppose that: a) \(\Gamma_1\subset\Gamma_2\); b) if \(b_1=(\zeta_{11}=x_1(\tau), a_{11}(x_1), N_{11},1)\), then the corresponding \(b_2\) has the form \(b_2=(\zeta_{2i}, a_{2i}(\xi), N_{2i},\delta\mu)\)

\[ (i=1,\ldots,s), \]

where \(\alpha\)) \(s\) and \(\|N_{2i}\|\) are bounded; \(\beta\))

\[ \sum_{\alpha}N_{2\alpha}=-N_{11}+\omega, \]

where \(\alpha\) are the indices corresponding to all \(\xi_{2\alpha}=\xi(0)\), \(\omega\to0\) as \(\varepsilon\to0\); c) the neighborhood \(O_{\zeta_{2i}}\) can be made arbitrarily small if one chooses a sufficiently small neighborhood \(O_{\zeta_{11}}\) and a suitable corresponding \(b_2\).

4.2. Now, as in case I, we shall consider the convex cone. Consider the trajectory and control \(\bar x(t)=x(t+(t_1'-\tau))\), \(\bar u(t)=u(t+(t_1'-\tau))\) of system (2.2).

Let \(x^*(t)\) be a variation of \(\bar x(t)\) corresponding to the symbol \(a\) and satisfying the condition \(x^*(t_1')-\bar x(t_1')=\varepsilon\delta\xi(0)\). Then we obtain

\[ x^*(T+t_1'-\tau+\varepsilon\delta t)-\bar x(T+t_1'-\tau)= \]

\[ =A_{T+t_1'-\tau,t}^{2}(\varepsilon\delta\xi(0))+\varepsilon\Delta\bar x_a+o(\varepsilon), \tag{4.5} \]

where \(A_{t,t'}^{2}\) is the transfer operator corresponding to system (2.2) along

\(\bar x(t),\ t'_1 \leqslant t \leqslant T+t'_1-\tau,\ x^*(t)\) is a variation from \(\bar x(t)\). On the other hand, from (4.2), (4.4) it follows that

\[ \delta \xi(0)=P^2_{0,\vartheta_1}(\delta\mu) \left( \frac{x^*_2(t_1)-x_1(t_1)+o(\varepsilon)}{\varepsilon} \right). \]

By virtue of assumption 3), a) with arbitrary \(b_1\), where \(\|N_{1i}\|\leqslant C\), we can choose \(b_2\) so that \(x^*_2(t_1)-x_1(t_1)=x^*_1(t_1)-x_1(t_1)=\varepsilon\delta'_2+o(\varepsilon)\) (see (4.3)), where \(x^*_1(t)\) is the corresponding \(N_1\)-variation from \(x_1(t)\). Then we obtain

\[ \varepsilon\delta\xi(0)=\varepsilon P^2_{0,\vartheta_1}(\delta\mu)\delta'_2+o(\varepsilon), \tag{4.6} \]

therefore, we can write (4.5) in the form

\[ x^*(T+t'_1-\tau+\varepsilon\delta t)-\bar x(T+t'_1-\tau)=\varepsilon\delta_3+o(\varepsilon) \tag{4.7} \]

with \(\delta_3=A^2_{T+t'_1-\tau,t'_1} P^2_{0,\vartheta_1}(\delta\mu)\delta'_2+\Delta\bar x_a\).

Note that when \(a,b\) vary, \(\delta_3\) generates a convex cone \(K^3\) with vertex \(\hat x(T)\). Let \(K\subset K'\times K^3\) be the set corresponding to \(\xi(0)=\delta x_1(\tau)\). It is easy to see that \(K\) is a convex cone. Then, as in case I, we can prove that \(K\) is separated from \(x_1\times L\), for otherwise \(x(t)\) and \(u(t)\) would not be optimal. Thus there exists a vector \(\bar\chi=(\bar\chi_1,\bar\chi_3)\ne 0\) such that

\[ (\bar\chi_1,\delta_1)+(\bar\chi_3,\delta_3)\leqslant 0. \tag{4.8} \]

Now consider the system

\[ \frac{d\bar\Psi}{dt} = -\frac{d\hat f_2(x(t),\bar u(t))}{dx}\,\bar\Psi . \tag{4.9} \]

Let \(\bar\Psi(t)\) be a solution satisfying the condition \(\bar\Psi(T+t'-\tau)=\bar\chi_3\). For empty \(a\) we have \((\bar\chi_3,\delta_3)=(\bar\Psi(t'_1),P^2_{0,\vartheta_1}(\delta\mu)\delta'_2)\). From (4.6) we obtain

\[ (\bar\chi_3,\delta_3)=(\bar\Psi(t'_1),\delta\xi(0))+o(\varepsilon). \tag{4.10} \]

Next we consider the system

\[ \frac{d\zeta}{d\theta} = \left( -\frac{\partial \hat f_2(x_2(t),u_2(t))}{\partial x} +\Lambda_2(\theta)\frac{\partial p_2(x_2(t),u_2(t))}{\partial x} \right)\zeta, \]

where \(\Lambda_2,\ p_2\) have the same meaning as in [1], with \(\hat f=\hat f_1,\ t=\theta-t'_1\). Let \(\xi(t)\) be the solution satisfying the condition

\[ \xi(0)=\bar\Psi(t'_1). \tag{4.11} \]

Then we obtain

\[ (\bar\Psi(t'_1),\delta\xi(0)) = (\xi(\vartheta_1),\delta^2_2) - \sum_{i=1}^{s} \left[ \lambda_2(0)a_{2i}(\xi(0)) \left( \frac{\partial g(x_2(t'_1))}{\partial x},N_{2i} \right)\delta\mu \right] - \]

\[ -\sum_{i=1}^{s}\int_{0}^{\vartheta_1} \left( \frac{d\lambda_2}{d\vartheta}a_{2i}(\xi(\vartheta)) \right) \left( \frac{\partial g(\xi(\vartheta))}{\partial x},N_{2i} \right) \delta\mu\,d\vartheta . \tag{4.12} \]

for \(\lambda_2(\vartheta)=(\xi(\vartheta),\Lambda_2(\vartheta))\). But from (4.3), (4.4) we obtain

\[ (\xi(\vartheta_1),\delta_2^2)=(\xi(\vartheta_1),\delta_2^1)+O(\varepsilon). \tag{4.13} \]

Now we consider the system

\[ \frac{d\Psi_1}{dt} = -\left( \frac{\partial \hat f_1(x_1(t),u_1(t))}{\partial x} + \Lambda_1(t)\frac{\partial p_1(x_1(t),u_1(t))}{\partial x} \right)\Psi_1, \]

where \(\Lambda_1, p_1\) have the same meaning as in [1], for \(\hat f=\hat f_1\). Let \(\Psi_1(t)\) be a solution of the system under consideration such that

\[ \Psi_1(t_1)=\xi(\vartheta_1). \tag{4.14} \]

Putting \(b_1=(\xi_{11}=x_1(\tau),a_{11}(x_1),N_{11},1)\), we have

\[ (\xi(\vartheta_1),\delta_2^1) = -(\Psi_1(\tau),N_{11}) + \lambda_1(\tau)\left(\frac{\partial g(x_1(\tau))}{\partial x},N_{11}\right) + \]

\[ + \int_\tau^t \frac{d\lambda_1}{dt} a_{11}(x_1(t)) \left( \frac{\partial g(x_1(t))}{\partial x},N_{11} \right)dt, \tag{4.15} \]

where \(\lambda_1(t)=-(\Psi_1(t),\Lambda_1(t))\). From (4.10), (4.12), (4.13), (4.15) it follows that

\[ (\bar\chi_3,\delta_3) = -(\Psi_1(\tau),N_{11}) + \lambda_1(\tau) \left( \frac{\partial g(x_1(\tau))}{\partial x},N_{11} \right) - \]

\[ - \sum_{i=1}^{s} \left[ \lambda_2(0)a_{2i}(\xi(0)) \left( \frac{\partial g(x_2(t_1'))}{\partial x},N_{2i} \right)\delta\mu \right] + \]

\[ + \int_\tau^{t_1} \frac{d\lambda_1}{dt} a_{11}(x_1(t)) \left( \frac{\partial g(x_1(t))}{\partial x},N_{11} \right)dt - \]

\[ - \sum_{i=1}^{s} \int_0^{\vartheta_1} \frac{d\lambda_2}{d\vartheta} a_{2i}(\xi(\vartheta)) \left( \frac{\partial g(\xi(\vartheta))}{\partial x},N_{2i} \right)\delta\mu\,d\vartheta + O(\varepsilon). \tag{4.16} \]

On the other hand, let \(\tilde\Psi(t)\) be a solution of system (2.9) satisfying the condition \(\tilde\Psi(0)=-\bar\chi_1\). Then we have

\[ (\bar\chi_1,\delta_1)=(\tilde\Psi(\tau),N_{11}) \tag{4.17} \]

for empty \(a\). Let \(b_2\) be the symbol corresponding to the symbol \(b_1\). Noting assumption 3), for sufficiently small \(\varepsilon\), \(O\varepsilon_{11}\), from (4.8), (4.16), (4.17) we have

\[ (\tilde\Psi(\tau)-\Psi_1(\tau)+\lambda_1(\tau)\operatorname{grad}g(x_1(\tau))+\lambda_2(0)\delta\mu\,\operatorname{grad}g(x_2(t_1')),N_{11})\le 0. \tag{4.18} \]

As \(\varepsilon\to0\) we have \(t_1'\to\tau\), \(\vartheta_1\to0\), \(\bar\chi\to\chi=(\chi_1,\chi_3)\ne0\) (since we can choose \(\|\bar\chi\|=1\)) and \(\bar x(t)\to x(t)\), \(\bar u(t)\to u(t)\) uniformly.

It follows from (4.9) that \(\widetilde{\Psi}(t)\) tends to the solution \(\Psi^+(t)\) of the system

\[ \frac{d\Psi^+}{dt} = \frac{-\,\partial \hat f_2(x(t),u(t))}{\partial x}\,\Psi^+ \]

under the condition \(\Psi^+(T)=\chi_3\). On the other hand, \(\widetilde{\Psi}(t)\) tends to the solution \(\Psi^-(t)\) of system (2.9) under the condition \(\Psi^-(0)=-\chi_1\). Further, from (4.11), (4.14) we obtain \(\Psi_1(\tau)\to\Psi^+(\tau)\), \(x_i(t)\to x(\tau)\) \((i=1,2)\), and from (4.18) we obtain

\[ \left( \Psi^-(\tau)-\Psi^+(\tau) + \bar\lambda_1(\tau) \left( \frac{\partial g(\hat x(\tau))}{\partial x} + \bar\lambda_2(0)\,\delta_\mu\, \frac{\partial g(\hat x(\tau))}{\partial x}, N_{11} \right) \right)\leq 0, \]

where \(\bar\lambda_1(\tau)\), \(\bar\lambda_2(0)\) are respectively the limits of \(\lambda_1(\tau)\), \(\lambda_2(0)\) as \(\varepsilon\to0\). In view of the arbitrariness of the vector \(N_{11}\), we obtain (2.16). It is easy to see that \(\Psi^-(t)\) and \(\Psi^+(t)\) satisfy conditions 1), 2) of the maximum principle [1]. Next, repeating the argument in case I, we obtain the jump conditions (2.16), (2.17), (2.18) in the preceding cases.

1. In conclusion, we can formulate the results obtained in the following theorem.

Theorem (on the jump condition). Suppose that conditions 1) and 2) are satisfied in case II and conditions 1), 2), 3) in case III. Then at the junction point \(\hat x(\tau)\) of the optimal trajectory one of the following conditions is fulfilled:

\[ \Psi^+(\tau)=\Psi^-(\tau)+\mu\,\operatorname{grad} g(\hat x(\tau)), \]

\[ \Psi^+(\tau)=\mu\,\operatorname{grad} g(\hat x(\tau)), \]

\[ \Psi^-(\tau)=-\mu\,\operatorname{grad} g(\hat x(\tau)),\quad \mu=\mathrm{const}, \]

where \(\Psi^-(t)\), \(\Psi^+(t)\) are the solutions, mentioned in the maximum principle, of the adjoint systems (2.9), (2.15) corresponding to systems (2.1), (2.2).

References

1. Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mishchenko E. F. Mathematical Theory of Optimal Processes. Moscow, 1961.

2. Gamkrelidze R. V. Izv. Akad. Nauk SSSR, Ser. Mat., 24, No. 3, 315–356, 1960.

Received by the editors
August 27, 1965

Hanoi State
University.
Democratic Republic
of Vietnam