H. Frankowska, "Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations," SIAM J. Control. Opt. 31 (1993), 257-272.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....system. Our treatment of V , although limited to the convex case, contrasts with other work on generalized Hamilton Jacobi equations which, in coping with discontinuities and values, has required H(x, y)tobe not just convex in y but also positively homogeneous in y; see Frankowska [7] [8], where is admitted directly, or more recently Bardi and Capuzzo Dolcetta [9; Chap. V, 5] where the conditions on H are narrower and is suppressed by nonlinear rescaling. Rescaling isn t compatible with an emphasis on convexity. While the interior of the set of points where V # could be ....

....because of the dominance of general subgradients in so much of the variational analysis and subdi#erential calculus in [6] The #V version would e#ectively turn (2. 13) into the one sided viscosity form of Hamilton Jacobi equation used for lsc functions by Barron and Jensen [12] and Frankowska [8], in distinction to earlier forms for continuous functions that called for pairs of inequalities, cf. Crandall, Evans and Lions [13] The book of Bardi and Capuzzo Dolcetta [9] gives a broad picture of viscosity theory in its current state, including the relationships between such di#erent forms. ....

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H. Frankowska, "Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations," SIAM J. Control. Opt. 31 (1993), 257-272.

....from inside. For the viscosity solution theory of first order Hamilton Jacobi equations, we refer to a new book by Bardi and Capuzzo Dolcetta [BC] On the other hand, in non smooth analysis, lsc solutions have been studied in optimal control theory. For the first result, we refer to Frankowska [F]. More recently, Wolenski and Zhuang [WZ] have proved the uniqueness of lsc solutions of the minimum time problem assuming the subsolution property on the target as in [S1] which the value function satisfies. We note that their definition of solutions is slightly different from that of viscosity ....

H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi equations, SIAM J. Control Optim., 31 (1993), 257-272.

....value functions are given in Proposition 2.5. The technique of treating semicontinuous solutions to the H J equation by using equivalence between the invariance property and the H J equation was first introduced by Subbotin [19] for differential games (see also Subbotin [20] and has been used in [8, 11] for finite horizon problems and in [22] for minimal time problems. The equivalence of the various concepts of the solution to the H J equation in an open set was also given in [8] We arrange the paper as follows: in the next section we state the problem formulation for the exit time problem and ....

H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, SIAM J. Contr. Optim. 31(1993), pp. 257-272.

....(# = 0) the answer to the above question is yes [18] Moreover, based on the new, nonsmooth versions of the verification theorems obtained, a scheme of obtaining feedback controls is proposed in [18] which does not involve any derivative of the value function. For some related works, see [2] [6], and [15] The present paper proceeds to answer the above question for stochastic systems. It should be noted that verification technique is particularly important for stochastic systems because only feedback controls perform well in the uncertain environment. However, the approach for the ....

H. FRANKOWSKA, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim., 31 (1993), pp. 257--272.

.... the specified initial conditions (t; x) x 2 K; and is the exit time of (s; x(s) from a closure of Omega 0 0 : The function g is defined as g(t; x) g(t; x) when (t; x) 2 [t 0 ; T ) Theta X g(x) when (t; x) 2 T Theta X (we can formally set g = 1 whenever x does not belong K [8]) This allows us to consider a family of optimization problems with different initial conditions (t; x) if we define the value function as the greatest lower bound of the functional (2.16) on the set of all admissible controls V (t; x) inf u fi 2 fi 1 2U J(t; x; u fi 2 fi 1 ) 2.17) ....

.... Sigma and Omega 0;1 0 : In this case the oscillation of the value function on the set U does not have to be defined explicitly 10 , and one can assume that the function V is merely a semi continuous function assuming its equality to 1 at all points from which the target K is unreachable [8, 36]. 10 if a function f has infinum m 0 and supremum M 0 on a set X; then the difference M 0 Gamma m 0 is called the oscillation of f on X In principle, the application of this idea is virtually unlimited [36] provided the Hamiltonian of the system is defined. However, the dilemma of this ....

Frankowska, H. Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, SIAM J. Control and Optimiz.,, vol.31, no.1, pp.257-272, 1993.

....(1.5) 3.1) 3.4) we can introduce the performance measure J(t; x t ; s t ) Z T t f 0 ( x ; s )d g(x(T ) 3.6) If we define the value function as V (t; x) def = inf s t 2UT J(t; x t ; s t ) 3. 7) then using appropriate regularity assumptions and dynamic programming principle [19, 20], the original optimal control problem can be studied through the Hamilton Jacobi Bellman (HJB) equation V (t; x t ) H(t;x t ; D x V (t; x t ) 0; V (T ; Delta) g( Delta) 3.8) where the Hamiltonian H is defined as H(t; x t ; ffi) def = sup s t 2UT f Gammaffi Delta f(t; x t ; s t ) ....

Frankowska, H., Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, SIAM J. Control and Optimiz.,, vol.31, no.1, pp.257-272, 1993.

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