The Pontryagin Maximum Principle and Transversality Conditions for a Class of Optimal Control Problems with Infinite Time Horizons

Abstract

Abstract. This paper suggests some further developments in the theory of first-order necessary optimality conditions for problems of optimal control with infinite time horizons. We describe an approximation technique involving auxiliary finite-horizon optimal control problems and use it to prove new versions of the Pontryagin maximum principle. Special attention is paid to the behavior of the adjoint variables and the Hamiltonian. Typical cases, in which standard transversality conditions hold at infinity, are described. Several significant earlier results are generalized.

An infinite-horizon maximum principle with bounds on the adjoint variable

“…In this section, we follow the general approach by Aseev et al (2001b) for the construction of necessary optimality conditions for (P) by considering a sequence of classical optimal control problems ðP k Þ; each defined on its own finite-time interval ½0; T k : Thereby T k pT kþ1 and T k ! 1 as k !…”

“…1: We will show that the sequence of optimal controls for ðP k Þ converges in L 2 to the optimal control for (P) entailing strong convergence of the corresponding state trajectories. Our problem (P) thereby does not satisfy the assumptions in Aseev et al (2001b), so that a number of modifications are necessary. Assume that ðx à ; u Ã Þ is an optimal pair for the original infinite-horizon optimal control problem (P), and let u ¼ maxfkuk : u 2 Ug: 8 Take a sequence of continuously differentiable functions v k : R þ !…”

“…The next proposition provides necessary optimality conditions for our problem (P), including bounds that imply the transversality condition and asymptotic stationarity of the Hamiltonian obtained by Aseev et al (2001b).…”

Infinite-horizon optimal advertising in a market for durable goods

“…In this section we follow the general approach by Aseev et al [23] and construct necessary optimality conditions for our problem (P) by considering a sequence of classical optimal control problems (P k ) where each is each defined on its own finite time interval ½0; T k ; where 05T k 4T kþ1 ; and T k ! 1 as k !…”

“…Using the method of smooth approximation (reviewed by Aseev [20]), it is possible to obtain weak transversality conditions in the form of asymptotic stationarity of the (maximized) Hamiltonian and positivity of the adjoint variables (see References [21][22][23]). Here, instead of imposing growth limitations and monotonicity on state trajectories, we modify the results in References [21][22][23] to suit our situation, where the state space is a compact invariant set, which in turn allows us to drop some restrictive assumptions on the evolution of the states. Under these natural conditions for our problem, we are able to obtain exponential bounds on the adjoint variables (i.e.…”