Asymptotic solution of a cheap control problem with state delay
The author notes that cheap controls (that is with small cost of control), with either point or distributed delays, were considered in several publications by R. E. O'Malley, P. Sanuti and others. Changes in a small cost of control amounts to a singular perturbation in the Hamilton-Jacobi-Bellman equation, and in the corresponding Hamilton boundary value problem. As the cost of control approaches zero, the limit of optimal trajectory corresponds to a singular arc. Thus the cheap control is a subclass of singularly perturbed control problems. The author comments that there are few works on singularly perturbed problems with delays, with the exception of his own technical reports (Technion, Israel, 1996). Let \(x(\tau)= \varphi_x(\tau)\), \(y(\tau)= \varphi_y(\tau)\) be the state variables, \(-h\leq \tau\leq 0\), while the state \[ \begin{multlined} dx/dt= A_1(t) x(t)+ A_2(t) y(t)+ H_1(t) x(t- h)+ H_2(t) y(t- h)+\\ \int^0_h [G_1(t, \tau) x(t+\tau)+ G_2(t, \tau) y(t+ \tau)] d\tau,\end{multlined} \] \[ \begin{multlined} dy/dt= A_3(t) x(t)+ A_4(t) y(t)+ H_3(t) x(t- h)+ H_4(t) y(t- h)+\\ \int^0_{-h} [G_3(t, \tau)x(t+ \tau)+ G_4(t, \tau)y(t+ \tau)] dt+ u(t),\end{multlined} \] \(t>0\), where \(x\) and \(y\) are vector variables, \(h\) is a positive constant \(A\), \(H\), \(G\) are matrices, while \(\varphi_i\) are continuous. The cost function is: \[ J_\varepsilon= \int^T_0 [x'(t) D_1(t) x(t)+ y'(t) D_2(t) y(t)+ \varepsilon^2u'(t) N(t) u(t)] dt, \] where primes denote transposition. \(D_1\) is a symmetric positive semidefinite matrix, \(D_2\) and \(N\) are symmetric positive definite matrices, \(\varepsilon\) is a small parameter. Using known results of V. B. Kolmanovskij and others, the author obtains the following results for optimal performance: \[ u^*(t)= -\varepsilon^2 N^{-1}(t) B'\left[P(t)z(t))+ \int^0_{-h} [Q(t, \tau) z(t+ \tau) d\tau]\right], \] where \(B\) is a matrix composed of an \(n\times m\) zero matrix and \(m\times m\) identity matrix. \[ J^*_\varepsilon= \varphi'(0) P(0)\varphi(0)+ 2\varphi' (0) \int^0_h Q(0,\tau)\varphi(t) d\tau+ \int^0_{-h} \int^0_{-h}\varphi'(\tau) R(0,\tau,\rho) \varphi(\rho) d\tau d\rho. \] Matrices \(P\), \(R\) and \(Q\) are unique solutions of the Riccati equations with zero conditions at \(t= T\), which arise in the usual manner in this control problem. The author proposes solutions which remove singularities. Then zero order asymptotic solutions are formally derived. Boundary layer corrections are computed in the neighborhood of \(t= T\) and \(\tau= -h\). The remainder of the paper contains proofs confirming validity of results which were derived by formal manipulation.
scientific article; zbMATH DE number 1389591