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Stochastic hybrid control. (English) Zbl 0967.93097
The authors consider a complicated version of controlled stochastic systems. The time $t$ is measured continuously. The state of the system is represented by a continuous variable $x$ and a discrete variable $n$. Also, the control has two parts, a continuous type control $v$ that is a measurable stochastic process and a discrete-type (or impulse) control $k$ that is a sequence of random variables. The key point is the set-interface $D$ of which only the boundary is really used. Minimal and maximal set-interfaces are considered. When the state reaches the minimal set, a mandatory impulse (jump or switch) takes place, while if the state belongs to a maximal set, an optional impulse (jump or switch) may be applied, upon decision of the controller. Switching and jumps can be autonomous or totally controlled. A discounted marginal cost of the form $f(x(t),n(t),v(t))\exp(-\int_{0}^{t}c(x(s),n(s),v(s)) ds)$ is introduced and a control problem consists in its minimization. The authors demonstrate that the dynamic programming approach leads to some involved quasi-variational inequality. If the system is non-degenerate then the classic treatment can be used for the solution of the control problem, otherwise, a way is to use the so-called viscosity solutions that are described in the last part of the paper.

##### MSC:
 93E20 Optimal stochastic control (systems) 93B12 Variable structure systems 49L25 Viscosity solutions (infinite-dimensional problems)
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##### References:
 [1] Antsaklis, P.; Stiver, J. A.; Lemmon, M. D.: Hybrid system modeling and autonomous control systems. Lecture notes in computer science, 366-392 (1993) [2] Aubin, J. P.: Impulse and hybrid control systems: A viability approach. (April 1999) [3] Back, A.; Guckenheimer, J.; Myers, M.: A dynamical simulation facility for hybrid systems. Lecture notes in computer science, 255-267 (1993) [4] Bainov, D. D.; Simeonov, P. S.: Systems with impulse effect. (1989) · Zbl 0671.34052 [5] Bellman, R. E.: Dynamic programming. (1957) · Zbl 0077.13605 [6] Bensoussan, A.: Stochastic control by functional analysis methods. (1982) · Zbl 0474.93002 [7] Bensoussan, A.; Lions, J. L.: Impulse control and quasi-variational inequalities. (1984) · Zbl 0324.49005 [8] Bensoussan, A.; Menaldi, J. L.: Hybrid control and dynamic programming. Dynam. continuous discrete impulsive syst. 3, 395-442 (1997) · Zbl 0897.49022 [9] Benveniste, A.; Le Guernic, P.: Hybrid dynamical systems theory and signal language. IEEE trans. Automat. control 35, 535-546 (1990) · Zbl 0709.68012 [10] Blankenship, G. L.; Menaldi, J. L.: Optimal stochastic scheduling of power generation systems with scheduling delays and large cost differentials. SIAM J. Control optim. 22, 121-132 (1984) · Zbl 0551.93076 [11] M. S. Branicky, Studies in Hybrid Systems: Modeling, Analysis, and Control, Thesis LIDS-TH-2304, MIT, Cambridge, MA, 1995. · Zbl 0874.68207 [12] Branicky, M. S.; Borkar, V. S.; Mitter, S. K.: A unified framework for hybrid control: model and optimal control theory. IEEE trans. Automat. control 43, 31-45 (1998) · Zbl 0951.93002 [13] Brockett, R.: Hybrid models for motion control systems. Essays in control: perspectives in the theory and its application, 29-53 (1993) [14] Costa, O. L. V.; Davis, M. H. A.: Impulse control of piecewise deterministic processes. Math. control signals syst. 2, 187-206 (1989) · Zbl 0675.93077 [15] Crandall, M. G.; Ishii, H.; Lions, P. L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. amer. Math. soc. 27, 1-67 (1992) · Zbl 0755.35015 [16] Davis, M. H. A.: Markov models and optimization. (1993) · Zbl 0780.60002 [17] Ezzine, J.; Haddad, A. H.: Controllability and observability of hybrid systems. Internat. J. Control 49, 2045-2055 (1989) · Zbl 0683.93011 [18] Fleming, W. H.; Soner, H. M.: Controlled Markov processes and viscosity solutions. (1992) · Zbl 0773.60070 [19] Hsu, C. S.: Cell-to-cell mapping. (1997) · Zbl 0875.93143 [20] Lions, P. L.: Generalized solutions of Hamilton--Jacobi equations. (1982) · Zbl 0497.35001 [21] Mariton, M.: Jump linear systems in automatic control. (1990) [22] Menaldi, J. L.: On the optimal impulse control problem for degenerate diffusions. SIAM J. Control optim. 18, 722-739 (1980) · Zbl 0462.93046 [23] Menaldi, J. L.: Optimal impulse control problems for degenerate diffusions with jumps. Acta appl. Math. 8, 165-198 (1987) · Zbl 0616.93079 [24] Nerode, A.; Kohn, W.: Models for hybrid systems: automata, topologies, sability. Lecture notes in computer science, 317-356 (1993) [25] Pontryagin, L. S.; Boltyanskii, V. C.; Gamkriledze, R. V.; Mischenko, E. F.: The mathematical theory of optimal processes. (1962) [26] M. Robin, Contröle impulsionnel des processus de Markov, Thesis TE-035, INRIA, Paris, France, 1977. · Zbl 0355.93039 [27] Tavernini, L.: Differential automata and their discrete simulators. Nonlinear anal. 11, 665-683 (1987) · Zbl 0666.34005 [28] Utkin, V. I.: Variable structure systems with sliding modes. IEEE trans. Automat. control 22, 212-222 (1977) · Zbl 0382.93036 [29] Varaiya, P. P.: Smart cars on smart mode: problem of control. IEEE trans. Automat. control 38, 195-207 (1993) [30] Witsenhausen, H.: A class of hybrid-state continuous-time dynamic systems. IEEE trans. Automat. control 12, 161-167 (1966)