Pontryagin's maximum principle in optimal control theory
The author investigates the Pontryagin's maximum principle which occured first time in the monograph ``The mathematical theory of optimal processes'' by \textit{L. S. Pontryagin, V. G. Boltyanskij, R. V. Gamkrelidze} and \textit{E. F. Mishchenko} (1961; Zbl 0102.31901). The author does not introduce any original results but brings together valuable views on this central result in the optimal control theory. He formulates the most general control problem: \(\dot x= f(x,u,t),\;t\in[t_1,t_2]\); \(u=(u_1,u_2)\), \(u_2(t)\in U_2(t)\) \(\dot\forall t,\) \(R(x,u_1,t)\leq 0\), \(G(x,t)\leq 0,\) \(K_1(p)\leq 0,\) \(K_2(p)=0\), \(p=(x_1,x_2,t_1,t_2)\), \(x_1=x(t_1)\), \(x_2=x(t_2),\) \(J=J(p,u)=K_0(p)+\int_{t_1}^{t_2} f^0(x,u,t) dt\to\min.\) The weak maximum principle is formulated firstly for the simplest problem \[ \dot x=f(x,u,t),\quad t\in[t_1,t_2];\quad u(t)\in U(t)\;\dot \forall t,\quad J(p,u)\to \min. \] The adjoint equation has the form \(\;\dot \psi = -\frac {\partial H}{\partial x}(t,\psi), \) the transversality conditions are \[ \psi(t_1^*)=\lambda^0\frac{\partial K_0}{\partial x_1}(p^*); \quad \psi(t_2^*)=-\lambda^0\frac{\partial K_0}{\partial x_2}(p^*), \] \[ \begin{aligned} \text{ess } \varlimsup_{t\to t_s^*} \sup_{u\in U(x_s^*,t)} & H\biggl( x_s^*,u,t,(-1)^{s-1}\lambda^0\frac {\partial K_0}{\partial x_s}(p^*)\biggl)+(-1)^{s-1}\lambda^0\frac {\partial K_0}{\partial t_s}(p^*)\geq 0;\\ \text{ess } \varliminf_{t\to t_s^*} \sup_{u\in U(x_s^*,t)} &H\biggl( x_s^*,u,t,(-1)^{s-1}\lambda^0\frac {\partial K_0}{\partial x_s}(p^*)\biggr)+(-1)^{s-1}\lambda^0\frac {\partial K_0}{\partial t_s}(p^*)\leq 0; \end{aligned} \] \(s=1,2.\) The condition of the maximum for the Pontryagin function \[ H(t,u^*(t))=\sup_{\substack{ u\in U(t)}} H(t,u) \] holds for almost every \(t\), where \[ H(t,u)=h(x^*(t),u,t,\psi(t),\lambda^0)=\langle f(x,u,t),\psi\rangle-\lambda^0f^0(x,u,t), \quad \lambda^0\geq 0. \] Two methods of proving the maximum principle are described. One is based on the penalty function method and is applied to linear convex problems. The second one is a combination of the finite-dimensional approximation technique and the penalty function method. Central attention is paid to the nondegeneracy conditions of the maximum principle for problems with phase constraints.
systems of ordinary differential equations
scientific article; zbMATH DE number 1348490
scientific article