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SIAM J. CONTROL OPTIM. Vol. 55, No. 2, pp. 1344–1345
ERRATUM: STOCHASTIC MINIMUM PRINCIPLE FOR PARTIALLY OBSERVED SYSTEMS SUBJECT TO CONTINUOUS AND JUMP DIFFUSION PROCESSES AND DRIVEN BY RELAXED CONTROLS∗ N. U. AHMED† AND CHARALAMBOS D. CHARALAMBOUS‡ Abstract. The goal of this note is to correct the gap in the proof of Lemma 3.1, p. 3239, in the paper [N. U. Ahmed and C. D. Charalambous, SIAM J. Control Optim., 51 (2013), pp. 3235–3257]. The gap was pointed out by Dr. Qingxin Meng, Department of Mathematics, Huzhou University, Zhejiang, China. Key words. stochastic differential equations, continuous diffusion, jump processes, relaxed controls, existence of optimal controls, necessary conditions of optimality AMS subject classifications. 49J55, 49K45, 93E20 DOI. 10.1137/16M1105189
1. Vague topology of relaxed controls. The set of admissible controls is given by Uad ≡ La∞ (I, M1 (U )), where U is a closed, bounded, possibly nonconvex subset of Rd . By Alaoglu’s theorem, this set endowed with the weak star (or vague) topology is compact. In the proof of Lemma 3.1 this was the basic assumption used. This assumption is too general (weak) for stochastic systems. It turns out that the following statement on page p. 3240 is incorrect: Now note that by virtue of vague convergence of un to uo the integrands of the above inequalities converge to zero for almost all s ∈ I, P -a.s.
For stochastic systems the vague topology is too weak for the continuity of the control to the solution map which Lemma 3.1 claims. A slightly stronger topology under which the above statement remains valid is given below. Consider the complete separable filtered probability space (Ω, F, Ft≥0 , P ) with Ft being right continuous having left limits, and let Gt≥0 be a nondecreasing family of subsigma algebras of the sigma algebra Ft≥0 . Let P denote the Gt≥0 predictable subsigma field of the product sigma field B(I)×F and µ the restriction of the product measure Leb. × P on it. Note that the measure space (I × Ω, P, µ) is separable and, since U is a compact metric space, the Banach space C(U ) with the standard supnorm topology is also separable. Thus the Lebesgue–Bochner space L1 (µ, C(U )) is a separable Banach space, and hence it has a countable dense set {ϕn }. Thus La∞ (I, M1 (U )), being a closed, bounded, convex subset of its dual, is metrizable. Using this fact, we can put several metric topologies on La∞ (I, M1 (U )). One suitable metric for our control problem is given by Z ∞ X d(u, v) ≡ (1/2n ) min{1, |ϕn (u)) − ϕn (v)|} dµ, n=1
I×Ω
∗ Received
by the editors November 28, 2016; accepted for publication December 9, 2016; published electronically April 27, 2017. http://www.siam.org/journals/sicon/55-2/M110518.html † Department of Mathematics, School of Engineering and Computer Science, University of Ottawa, Ottawa, ON, K1N 6N5, Canada (
[email protected]). ‡ Department of Electrical and Computer Engineering, University of Cyprus, Nicosia, Cyprus (
[email protected]). 1344
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Downloaded 09/06/18 to 139.81.241.129. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
ERRATUM
1345
R where ϕn (u)(t, ω) ≡ U ϕn (t, ω, ξ)ut,ω (dξ). This is a complete metric space. We denote this metric space by (M, d) and take any compact subset Uad of this metric space as the set of admissible controls. Convergence in this metric topology implies that, for any ϕ ∈ L1 (µ, C(U )), there exists a subsequence of the sequence {ϕ(un )} that converges to ϕ(uo )-µ a.e. With respect to this set of admissible controls, Lemma 3.1 remains valid. REFERENCE [1] N. U. Ahmed and C. D. Charalambous, Stochastic minimum principle for partially observed systems subject to continuous and jump diffusion processes and driven by relaxed controls, SIAM J. Control Optim., 51 (2013), pp. 3235–3257, https://doi.org/10.1137/120885656.
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