By Krotov V.

This paintings describes all easy equaitons and inequalities that shape the mandatory and enough optimality stipulations of variational calculus and the speculation of optimum regulate. topics addressed comprise advancements within the research of optimality stipulations, new sessions of ideas, analytical and computation equipment, and purposes.

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Show that if u is a subsolution of a proper equation F (u, Du, D 2 u) = 0, on IRN , then ft is as well. 6), show 12— 2 <2k((2) — Conclude that if u is a bounded subsolution of an equation F (u, Du, D 2 u) = f (x) f (x) + where f is uniformly continuous, then 'a is a solution of F(û, Dû, D 2 û) (5, for some constant 6. , —> 0 as n O. Discuss the general case, F(x,Du,D 2 u) < O. * in place of the Theorem on Sums while working on problems in this area. We will also employ two nontrivial facts about semiconvex functions.

33] K. Miller, Barriers on cones for uniformly elliptic equations, Ann. di Mat. Pura Appl. LXXVI (1967), 93-106. [34] M. Soner, Controlled Markov processes, viscosity solutions and applications to mathematical finance, this volume. 43 [35] P. E. Souganidis, Front Propagation: Theory and applications, this volume. [36] A. Subbotin, Solutions of First-order PDEs. The Dynamical Optimization Perspective, Birkhauser, Boston, 1995. [37] A. wiçch, W"-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, preprint.

I Here is another useful form of the DPP, which is closer to Bellman's original Principle of Optimality. 2. For all a(-) E A the following function is nondecreasing: s Infinite Horizon : s 1-> f gyz (t),a(t))e -t dt + V(y z (s,a))e- 8 , s E [0, +ook o Finite Horizon : s i- v(y z (s,a),t - s), s E [0, t]; Minimum Time : s 1-* s + T(yz (s,a)), s E [0,4(a)], if T(x) <+00; s Discounted Minimum Time : s }--4 f e -t dt + V(yz (s,a))e', s E [0, tz (a)]. o Moreover this function is constant if and only if the control a(-) is optimal for the initial position x (and the horizon t in the finite horizon problem).