Bounds for a risk-sensitive homing problem. (English) Zbl 1539.93079
Summary: Under extra conditions, we establish some new bounds for a risk-sensitive homing problem with control only in the diffusion-coefficient of a controlled diffusion process in a given closed interval. Our main results generalize or strengthen those in [M. Lefebvre, Ann. Appl. Probab. 14, No. 2, 786–795 (2004; Zbl 1061.93099)], and a recent result proved by the present author [IEEE Trans. Autom. Control 67, No. 7, 3770–3772 (2022; Zbl 1537.93697)].
MSC:
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
References:
| [1] | Bihari, I., A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Mathematica Academiae Scientiarum Hungaricae, 7, 81-94, 1956 · Zbl 0070.08201 |
| [2] | Enhao, Y., On some nonlinear integral and discrete inequalities related to Ou-Iang’s inequality, Acta Mathematica Sinica, 14, 353-360, 1998 · Zbl 0913.26007 |
| [3] | Kuhn, J., The risk-sensitive homing problem, Journal of Applied Probability, 22, 796-803, 1985 · Zbl 0607.93065 |
| [4] | Lefebvre, M., LQG homing in two dimensions, IEEE Transactions on Automatic Control, 32, 639-641, 1987 · Zbl 0618.93075 |
| [5] | Lefebvre, M., An extension of a stochastic control theorem due to Whittle, IEEE Transactions on Automatic Control, 34, 567-568, 1989 |
| [6] | Lefebvre, M., Risk-averse control of a three-dimensional wear process, International Journal of Control, 73, 1307-1311, 2000 · Zbl 0993.90040 |
| [7] | Lefebvre, M., A homing problem for diffusion processes with control-dependent variance, Annals of Applied Probability, 14, 786-795, 2004 · Zbl 1061.93099 |
| [8] | Makasu, C., Risk-sensitive control for a class of homing problems, Automatica, 45, 2454-2455, 2009 · Zbl 1183.93128 |
| [9] | Makasu, C., Homing problems with control in the diffusion coefficient, IEEE Transactions on Automatic Control, 67, 3770-3772, 2022 · Zbl 1537.93697 |
| [10] | Øksendal, B., Stochastic differential equations: An introduction with applications, 2003, Springer-Verlag: Springer-Verlag Berlin · Zbl 1025.60026 |
| [11] | Ou-Iang, L., The boundedness of solutions of linear differential equations \(y'' + A ( t ) y = 0\), Shuxue Jinzhan, 3, 409-415, 1957 |
| [12] | Pachpatte, B. G., On a certain inequality arising in the theory of differential equations, Journal of Mathematical Analysis and Applications, 182, 143-157, 1994 · Zbl 0806.26009 |
| [13] | Whittle, P., Optimization over time. Vol. 1, 1982, Wiley: Wiley Chichester · Zbl 0557.93001 |
| [14] | Whittle, P., Risk-sensitive optimal control, 1990, Wiley: Wiley Chichester · Zbl 0718.93068 |
| [15] | Willet, D.; Wong, J. S.W., On the discrete analogues of some generalizations of Grönwall’s inequality, Monatsh für Mathematik, 69, 362-367, 1965 · Zbl 0145.06003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
