Most Rapid Approach Paths And Applications

Many infinite-horizon optimal problems encounter in management scienceand economics have solutions that approach some stationary value for the state variable. Spence and Starret [2] have delineated a class of problemin which the optimal solution is to approach an optimal stationary level, when it exists, as fast as possible. They term such paths as most rapid approach paths.There are many possible reasons for being interested in such problems. Compared with other types of solutions, they are relatively simple. They are easy to describe, and to compute, and perhaps easier to explain to the non-technical person than other kinds of solutions. Furthermore, most rapid approach policies seem to accord with the layman's intution as to what should be done a number of interesting economic problems, including polution and self-renewing natural resources.In the preface, we introduced some proposition about the existence of the most rapid approach paths in the continuous-time and discrete-time problem.Propostion 1 If the derives utility functions for problem (1.1) take the form (1.2), then the problem can be transformed into the formsubject to ??? = G(x,u),x(0) = x₀,u∈U(x), U(x) compact and connected.Proposition 2 For problem (1.3)V satisfying P.1, the optimal path has the property that it approachs as possible some x which locally maximizesV, and then stays there forever.The most rapid approach theorem does hold in discrete time for quasiconcaveV,Proposition 3 If V(x) is quasi-convex and is maximized x, and if the most rapid approach path from x₀ to x is well-defined and feasible, then it is an optimal path.In economics, most rapid approach path applied in self-renewing naturalresource,water pollution and human capital.In 1977, S.P.SETHI propose using Green's theorem to solve the problemof most rapid approach paths. [5], Consider the optimal control problem with one state and one control varaiable:with the discount r > 0, subject toS.P.SETHI gave the following theorem [5]:Theorem 4 Let the following conditions hold: (i) x is the unique solution of (2.15);(ii) u is the unique solution of f{(?),(?)) = 0, i.e. (?) = u((?),0) as defined in(2.4); (iii) (?)(iv) (?) is feasible;(v) there is a most rapid approach path from x₀ to (?); (vi) T is sufficiently large, so that, after attainingx using (iv), we still have enough time to attain x_t Then, the most rapid approach paths from x₀ to (?), followed by a stay at (?), followed by an exit from (?) as late as possible to attain x_t at time T is optimal.where I(x) = -[rN(x) + M'(x)].In 1987, R. F. HARTAL and G. FEICHTIGER considered nonautonomousoptimal control problem [6], and gave the following theorem:Theorem 5 Assume that, for every t≥0, (2.26) has a unique solution (?)(t) and that x is feasible, i.e.,Furthmore, for t≥0, letFinally, we assume that, for any feasible trajectory x(t),If there exists a most rapid approach path from X₀ to (?)(t), then this trajectoryis optimal.where I(x, t) = -rN((?), t) + N_t((?), t) - M_x((?), t).PAPAPORT A. and CARTICNY P. respectly proposed using an appraochwith the value funcion to solve most rapid approach paths problem for one varaible where the velocity bounded in 2002 and 2005 [3] [4]. we consider a problem of the calculus of variations in infinite horizon and n variables whose objective is given bywhereδ> 0 and l(·) is a real valued function on Rⁿ×Rⁿ linear w.r.t. the derivative.where x(·) = (x₁(·),…,x_n(·)), f^-(·) = (f₁^-(·),…,f_n^-(·)),We have the following important results:Proposition 6 We assume Hypotheses (H₁)-(H₄). V₂(·) is a continuousviscosity on (a, b) of the following Hamilton-Jacobi equation:Proposition 7 Under Hypotheses (H1)-(H6), the following statementsare equivalent: (i) T{x) > 0, (?)x∈(a, b)ⁿ.(ii) (?) = (a,b)ⁿ and b(?) = {x∈(a, b)ⁿ|T(x) = S(x, (?))}, (?).Corollary 8 Let Hypotheses (H1)-(H6) hold.(i) If the function h(·) has one exactly one zero (?) on (a, b)×…×(a, b) and fulfills the following property at any x∈(a, b)ⁿ, then the most rapid approach path to (?) is optimal from, any initial condition x₀∈(a, b)ⁿ.(ii) For any (?) such that B(?) is nonempty, there exists a neighborhoodV of (?), such that the property (9) is fulfilled on V.