Article PDF
Avoid common mistakes on your manuscript.
References
R. A. Adiatullina and A. M. Tarasyev, “An infinite time horizon differential game,” Prikl. Mat. Mekh., 51, 531–537 (1987).
L. D. Akulenko, Asymptotic Methods in Optimal Control [in Russian], Nauka, Moscow (1987).
V. M. Alexeev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control [in Russian], Nauka, Moscow (1979).
M. I. Alekseichik, “An advanced formalization of the basic notions to antagonistic differential games,” Mat. Anal. Prilozh., Rostov-on-Don State Univ., 7, 191–199 (1975).
E. H. Albrecht, “Constructions of approximate solutions to quasi-linear differential games,” Proc. Steklov Inst. Math., Suppl. Issue 1, S24–S34 (2000).
E. H. Albrecht and G. S. Shelement’ev, Lectures on the Theory of Stabilization [in Russian], Ural State Univ., Sverdlovsk (1972).
B. I. Ananiev, “On minimax state estimates for multistage statistically uncertain systems,” Probl. Contr. Inform. Theory, 18, No. 1, 27–41 (1981).
V. I. Arnold, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1974).
V. I. Arnold, Singularities of Caustics and Wave Fronts [in Russian], Fazis, Moscow (1996).
Z. Artstein and V. Gaitsgory, “Tracking fast trajectories along a slow dynamics. A singular perturbations approach,” SIAM J. Contr. Optimiz., 35, 1487–1507 (1997).
Z. Artstein and V. Gaitsgory, “The value function of singularly perturbed control systems,” Appl. Math. Optimiz., 41, 425–445 (2000).
A. V. Arutyunov and S. M. Aseev, “The maximum principle in differential inclusions with phase restrictions,” Dokl. Ross. Akad. Nauk, 334, No. 2, 134–137 (1994).
J.-P. Aubin, Viability Theory, Birkhäuser, Boston (1991).
J.-P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin—Heidelberg (1984).
J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis [Russian translation], Mir, Moscow (1988).
J.-P. Aubin and H. Frankowska, Set Valued Analysis, Birkhäuser, Boston, (1990)
N. S. Bakhvalov, Average of Processes in Periodic Media [in Russian], Nauka, Moscow (1984).
A. E. Barabanov and A. M. Ghulchak, “H-infinity optimisation problem with sign-indefinite quadratic form,” Systems Control Lett., 29, 157–164 (1996).
N. E. Barabanov and R. Ortega, “Necessary and sufficient conditions for passivity of the Lugre friction model,” IEEE Trans. Autom. Contr., 1 (2000).
E. A. Barbashin, Introduction to the Theory of Stability [in Russian], Nauka, Moskow (1967).
M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton—Jacobi—Bellman Equations, Birkhäuser, Boston (1997).
M. Bardi and L. C. Evans, “On Hopf’s formulas for solutions of Hamilton—Jacobi equations,” Nonlinear Analysis, Theory, Methods, Appl., 8, No. 11, 1373–1381 (1984).
M. Bardi and M. Falcone, “An approximation scheme for the minimum time function,” SIAM J. Control Optimiz., 28, 950–965 (1990).
G. Barles and B. Perthame, “Exit time problems in optimal control and vanishing viscosity solutions of Hamilton—Jacobi equations,” SIAM J. Control Optimiz., 26, 1133–1148 (1988).
E. N. Barron and R. Jensen, “The Pontryagin maximum principle from dynamical programming and viscosity solutions to first-order partial differential equations,” Trans. Amer. Math. Soc., 298, No. 2, 635–641 (1986).
E. N. Barron, L. C. Evans, and R. Jensen, “Viscosity solutions of Isaacs’ equations and differential games with Lipschitz controls,” J. Differ. Equations, 53, 213–233 (1984).
T. Basar and P. Bernhard, H ∞-Optimal Control and Related Minimax Design Problems, Birkhäuser, Boston (1991).
V. D. Batukhtin, “The extremal aiming in a nonlinear approach game,” Dokl. Ross. Akad. Nauk, 27, No. 1 (1972).
R. Bellman, Dynamic Programming, Princeton Univ. Press, Princeton, New Jersey (1957).
R. Bellman and R. Kalaba,Dynamic Programming and Modern Control Theory, Academic Press, New York (1965).
A. Bensoussan and J. L. Lions, Applications of Variational Inequalities in Stochastic Control, North-Holland, Amsterdam—New York—Oxford (1982).
A. Bensoussan, Perturbation Methods in Optimal Control, Wiley-Gauthier, New York—Chichester (1988).
Yu. I. Berdyshev, “A qualitative analysis of attainability sets,” Kosmich. Issled., 34, No. 2, 141–144 (1996).
L. D. Berkovitz, “A variational approach to differential games,” in: Adv. Game Theory, Ann. Math. Stud., 52, Princeton Univ. Press, Princeton (1964).
L. D. Berkovitz, “Optimal feedback controls,” SIAM J. Control Optimiz., 27, 991–1006 (1989).
V. I. Blagodatskikh and A. F. Filippov, “Differential inclusions and optimal control problems,” Proc. Steklov Inst. Math., 169, 194–252 (1985).
N. D. Botkin, M. A. Zarkh, V. N. Kein, V. S. Patsko, and V. L. Turova, “Differential games and control problems for an aircraft under windshear,” Izv. Ross. Akad. Nauk. Ser. Tekhn. Kibern., No. 1, 68–76 (1993).
V. G. Boltyanskii, Mathematical Methods of Optimal Control [in Russian], Nauka, Moscow (1966).
A. Bryson and Y.-C. Ho, Applied Optimal Control Theory [Russian translation], Mir, Moscow (1972).
S. A. Brykalov, “A conflict controlled system with a nonfixed end time moment,” Proc. Steklov Inst. Math., Suppl. Issue 2, S313–S319 (2000).
R. Bulirsh, E. Ners, H. J. Pesh, and O. von Stryk, Combining direct and indirect methods in optimal control: angle maximisation of a hang glider, Report Shcwerpunktprogr. DFG Anwendungsabgene Optimierung und Steuerung, No. 313, Math. Inst. Techn. Univ. München (1991).
P. Cannarsa and H. Frankowska, “Some characterization of optimal trajectories in control theory,” SIAM J. Control Optimiz., 29, 1322–1347 (1991).
A. G. Chentsov, “On an approach game problem at a given time moment,” Mat. Sb., 99, 394–420 (1976).
A. G. Chentsov, Asymptotic Attainability, Kluwer Academic, Dordrecht (1997).
A. G. Chentsov and V. E. Pak, “On the extension of the nonlinear problem of optimal control with nonstationary phase restrictions,” Nonlinear Analysis: Theory, Methods, Appl. 26, No. 2, 383–394 (1996).
F. L. Chernous’ko, L. D. Akulenko, and B. N. Sokolov, Control of Oscillations [in Russian], Nauka, Moscow (1980).
F. L. Chernous’ko and V. B. Kolmanovskii, Optimal Control under Random Disturbances [in Russian], Nauka, Moscow (1978).
F. L. Chernous’ko and A. A. Melikyan, Game Problems of Control and Search [in Russian], Nauka, Moscow (1978).
A. A. Chikrii, Conflict Controlled Processes [in Russian], Naukova Dumka, Kiev (1992).
S. V. Chistyakov, “On solutions of pursuit game problems,” Prikl. Mat. Mekh., 41, 825–832 (1977).
F. H. Clarke, “Generalized gradients and applications,” Trans. Amer. Math. Soc., 205, 246–262 (1975).
F. H. Clarke, Optimization and Nonsmooth Analysis [Russian translation], Nauka, Moscow (1988).
F. H. Clarke and Yu. S. Ledyaev, “New formulas for finite differences,” Dokl. Ross. Akad. Nauk, 331, No. 3, 275–277 (1962).
F. H. Clarke, Yu. S. Ledyaev, and R. Stern, “Proximal analysis and feedback constructions,” Proc. Steklov Inst. Math., Suppl. Issue 1, S72–S89 (2000).
F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York (1997).
F. H. Clarke and R. Vinter, “The relationship between the maximum principle and dynamic programming,” SIAM J. Contr. Optimiz. No. 5, 1291–1311 (1987).
E. D. Conway and E. Hopf, “Hamilton’s theory and generalized solutions of the Hamilton—Jacobi equations,” Trans. Amer. Math. Soc., 13, No. 2, 939–986 (1964).
M. G. Crandall, “A generalization of Peano’s existence theorem and flow invariance,” Proc. Amer. Math. Soc., 36, No. 1, 151–155 (1972).
M. G. Crandall and P. L. Lions, “Viscosity solutions of Hamilton—Jacobi equations,” Trans. Amer. Math. Soc., 277, 1–42 (1983).
M. G. Crandall, L. C. Evans, and P. L. Lions, “Some properties of viscosity solutions of Hamilton—Jacobi equations,” Trans. Amer. Math. Soc., 282, 487–502 (1984).
M. G. Crandall, H. Ishii, and P. L. Lions, “A user’s guide to viscosity solutions,” Bull. Amer. Math. Soc., 27, 1–67 (1992).
R. Courant and D. Hilbert, Partial Differential Equations [Russian translation], Mir, Moscow (1964).
A. R. Danilin, “Asymptotics of controls to a singular elliptic problem,” Dokl. Ross. Akad. Nauk, 369, No. 3, 305–308 (1999).
V. F. Demyanov and V. N. Malozemov, An Introduction to Minimax [in Russian], Nauka, Moscow (1972).
V. F. Demyanov and A. M. Rubinov. Foundations of Nonsmooth Analysis and Quasi-differential Calculus [in Russian], Nauka, Moscow (1990).
V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Peter Lang, Frankfurt (1995).
M. G. Dmitriev, “The theory of singular perturbations and some problems of optimal control,” Differ. Equations, 21, No. 10, 1693–1698 (1985).
A. Donchev, Optimal Control Systems. Perturbations, Approximations and Analysis of Sensitivity [Russian translation], Mir, Moscow (1987).
A. Ya. Dubovitzkii and A. A. Milyutin, “Problems on extremum under restrictions,” Dokl. Akad. Nauk SSSR, 149, No. 4, 759–762 (1963).
V. Ya. Dzhafarov, “On stability of the guaranteed result to a positional control problem,” Dokl. Acad. Nauk SSSR, 285, No. 1, 27–31 (1985).
R. Elliott, “Viscosity solutions and optimal control,” in: Pitman Res. Notes, Math. Ser., 165, Boston (1987).
R. J. Elliott and N. J. Kalton, “The existence of value in differential games of pursuit and evasion,” J. Differ. Equations, 12, No. 3, 504–523 (1972).
L. C. Evans, Partial Differential Equations, Grad. Stud. Math., 19, Amer. Math. Soc., Providence, Rhode Island (1998).
A. F. Filippov, “On some questions in the theory of optimal control,” Vestn. Moscow State Univ. Ser. Math., Mech., Phys., Chem., No. 2, 25–32 (1959).
A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Kluwer Academic, Dordrecht (1988).
W. H. Fleming, “The convergence problem for differential game, II,” Adv. Game Theory, Ann. Math. Stud., 52, 195–210 (1964).
W. H. Fleming, “The Cauchy problem for degenerate parabolic equations,” J. Math. Mech., 13, No. 6, 987–1008 (1964).
W. H. Fleming, “The Cauchy problem for a nonlinear first order differential equation,” J. Differ. Equations, 5, No. 3, 515–550 (1969).
W. H. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control [Russian translation], Mir, Moscow (1978).
W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York (1993).
V. N. Fomin, A. M. Fradkov, and V. A. Yakubovich, Adaptive Control of Dynamical Objects [in Russian], Nauka, Moscow (1981).
A. Friedman, Differential Games, Wiley Interscience, New York (1971).
R. F. Gabasov and F. M. Kirillova, Qualitative Theory of Optimal Processes [in Russian], Nauka, Moscow (1971).
V. G. Gaitsgory, Control of Systems with Fast and Slow Motions [in Russian], Nauka, Moscow (1991).
V. G. Gaitsgory, “Suboptimal control of singularly perturbed systems and periodic optimization,” IEEE Trans. Automat. Contr., 38, No. 6, 888–902 (1993).
V. G. Gaitsgory, “Limit Hamilton—Jacobi equations for singularly perturbed zero-sum differential games,” J. Math. Anal. Appl., 202, 862–899 (1996).
R. V. Gamkrelidze, Principles of Optimal Control [in Russian] Tbilisi State Univ. (1975).
R. V. Gamkrelidze, A. A. Agrachev, and S. A. Vakhrameev, “Ordinary differential equations on vector fibers and chronological calculus,” in: Itogi Nauki i Tekhn. Ser. Sovr. Probl. Mat., 35, All-Union Institute for Scientific and Technical Information, Moscow (1989), pp. 3–107.
V. L. Gasilov and V. B. Kostousov, “Methods of an extraction and a representation of information about geophysical fields,” Gyroscopes Navigation, 4, No. 15, 64–65 (1996).
I. M. Gel’fand, “Some problems to the theory of quasi-linear equations,” Usp. Mat. Nauk, 14, No. 2, 87–158 (1959).
E. I. Gerashchenko and S. M. Gerashchenko, Method of Decomposition of Motions and Optimization of Nonlinear Systems [in Russian], Nauka, Moscow (1975).
S. K. Godunov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1979).
A. Yu. Goritski and E. Yu. Panov, “Example of nonuniqueness of entropy solutions in the class of locally bounded functions,” Zh. Mat. Fiz., 6, No. 4, 492–494 (1999).
N. L. Grigorenko, Yu. N. Kiselev, N. V. Lagunov, D. B. Silin, and N. G. Trin’ko, Methods of Solving Differential Games. Mathematical Modelling [in Russian], Moscow State Univ. (1993).
S. V. Grigoryeva, A. M. Tarasyev, A. A. Uspenskii, and V. N. Ushakov, “Constructions of the theory of differential games to solving the Hamilton—Jacobi equations,” Proc. Steklov Inst. Math., Suppl. Issue 2, S320–S336 (2000).
M. I. Gusev and A. B. Kurzhanskii, “On optimization of controlled systems with restrictions, I, II,” Differ. Equations, 7, No. 9, 1591–1602; No. 10, 1789–1800 (1971).
M. I. Gusev, “On the structure of optimal minimax estimates to problems of the guaranteed estimate,” Dokl. Ross. Akad. Nauk, 322, No. 5, 832–835 (1992).
Kh. G. Guseinov and V. Ya. Dzhafarov, “Left-hand side solutions of the Hamilton-Jacobi equations,” Proc. Steklov Inst. Math., Suppl. Issue 2, S337–S350 (2000).
Kh. G. Guseinov and V. N. Ushakov, “On constructions of differential inclusions with given properties,” Differ. Equations, 36, No. 4, 438–445 (2000).
Kh. G. Guseinov, A. I. Subbotin, and V. N. Ushakov, “Derivatives of multivalued mappings and their applications to game-theoretic control problems,” Probl. Control Inform. Theory, 14, No. 3, 1–14 (1985).
P. B. Gusyatnikov, Theory of Differential Games [in Russian], Moscow Phys.-Tekhn. Inst. (1982).
G. Haddad, “Monotone trajectories of differential inclusions and functional-differential inclusions with memory,” Isr. J. Math., 39, 83–100 (1981).
C. J. Himmelberg, “Measurable relations,” Fundam. Math., 82, No. 1, 53–72 (1975).
E. Hopf, “Generalized solutions of nonlinear equations of first order,” J. Math. Mech., 14, 951–972 (1965).
A. M. Il’in, “Boundary layers,” in: Itogi Nauki i Tekhn. Ser. Fundam. Napr., 34, All-Union Institute for Scientific and Technical Information, Moscow (1988).
A. M. Il’in, A. S. Kalashnikov, and O. A. Oleinik, “Second-order linear equations of parabolic type,” Usp. Mat. Nauk, 17, No. 3, 3–146 (1962).
A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems [in Russian], Nauka, Moscow (1974).
R. Isaacs, Differential Games, Wiley, New York (1965).
H. Ishii and S. Koike, “Remarks on elliptic singular perturbation problems,” Appl. Math. Optim., 23, 1–75 (1991).
A. Isidori, Nonlinear Control Systems, Springer-Verlag, New York (1995).
A. G. Ivanov, “The problem on brachistocrone in the central gravitation field,” in: Algorithms and Software of Parallel Calculations [in Russian], 2, Ekaterinburg (1998), pp. 95–109.
E. Kamke, Handbook on First-Order Partial Differential Equations [Russian translation], Nauka, Moscow (1966).
L. V. Kamneva, V. S. Patsko, and V. L. Turova, “Construction of the value function for game brachistochrone problem,” in: Proc. X Int. Symp. “Dynamical Games and Applications,” July, 8–11, 2002, St. Petersburg State University, Russia, 2, St. Petersburg (2002), pp. 408–415.
L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977).
I. Ya. Katz, Method of Lyapunov Functions in Problems of Stability and Stabilization for Systems of Random Structure [in Russian], Ekaterinburg (1998).
M. M. Khrustalev, “Necessary and sufficient optimality conditions in terms of the Bellman equation,” Dokl. Akad. Nauk SSSR, 254, 293–297 (1980).
A. V. Kim and V. G. Pimenov, “On applications of i-smooth analysis to numerical methods of solving finctional-differential equations,” in: Proc. Inst. Math. Mechanics, Ural Branch of the Russian Academy of Sciences, 5, Ekaterinburg (1998), pp. 119–142.
A. F. Kleimenov, Nonantagonistic Positional Differential Games [in Russian], Nauka, Ekaterinburg (1993).
A. I. Klimushev and N. N. Krasovskii, “Uniform asymptotic stability of systems of ordinary differential equations with a small parameter at derivatives,” Prikl. Mat. Mekh., 25, 1011–1025 (1962).
P. V. Kokotovic, “Applications of singular perturbations techniques to control problems,” SIAM J. Rev., 26, No. 4, 501–510 (1984).
A. N. Kolmogorov and S. V. Fomin, Elements of Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1968).
V. N. Kolokol’tsov and V. P. Maslov, Idempotent Analysis and Applications to Optimal Control Problems [in Russian], Nauka, Moscow (1994).
A. F. Kononenko, “On structure of an optimal strategy to dynamical control systems,” Zh. Vychisl. Mat. Mat. Fiz., 20, No. 5, 1105–1116 (1980).
A. I. Korotkii, “Dynamical modelling of parameters to hyperbolic systems,” Izv. Akad. Nauk SSSR, Ser. Tekhn. Kibern., No. 2, 154–164 (1991).
A. N. Krasovskii, “A differential game for the positional functional,” Dokl. Akad. Nauk SSSR, 253, No. 6, 1303–1307 (1980).
A. N. Krasovskii and N. N. Krasovskii, Control under Lack of Information, Birkhäuser, Boston (1995).
N. N. Krasovskii, Theory of Control of Motions [in Russian], Nauka, Moscow (1968).
N. N. Krasovskii, Game-Theoretic Problems on the Encounter of Motions [in Russian], Nauka, Moscow (1970).
N. N. Krasovskii and N. Yu. Lukoyanov, “Equations of the Hamilton—Jacobi type in hereditary systems: Minimax solutions,” Proc. Steklov Inst. Math., Suppl. Issue 1, S136–S154 (2000).
N. N. Krasovskii and V. M. Reshetov, “Approach-evasion problems for systems with a small parameter at derivatives,” Prikl. Mat. Mekh., 38, No. 5, 771–779 (1974).
N. N. Krasovskii and T. N. Reshetova, “On the program synthesis of a guaranteed control,” Probl. Contr. Inform. Theory, 17, No. 6, 333–343 (1988).
N. N. Krasovskii and A. I. Subbotin, “An alternative to an approach game problem,” Prikl. Mat. Mekh., 34, 1005–1022 (1970).
N. N. Krasovskii and A. I. Subbotin, Positional Differential Games [in Russian], Nauka, Moscow (1974).
N. N. Krasovskii, Control of Dynamical Systems [in Russian], Nauka, Moscow (1985).
N. N. Krasovskii and A. I. Subbotin, Game-Theoretical Control Problems, Springer-Verlag, New York (1988).
N. N. Krasovskii and V. E. Tretyakov, “A saddle point of a stochastic differential game,” Dokl. Akad. Nauk SSSR, 254, No. 3, 534–539 (1980).
V. F. Krotov, “Global methods in optimal control theory,” in: Advances in Nonlinear Dynamics and Control: A Report from Russia, Progr. Systems =Control Theory, No. 17, Birkhäuser, Boston (1993), pp. 76–121.
V. F. Krotov and V. I. Gurman, Methods and Problems of Optimal Control [in Russian] Nauka, Moscow (1973).
S. N. Kruzhkov, “On methods of construction of generalized solutions of the Cauchy problem for a quasi-linear partial differential equation of first order,” Usp. Mat. Nauk, 20, No. 6, 112–118 (1965).
S. N. Kruzhkov, “Generalized solutions of first-order nonlinear equations in several independent variables, I,” Mat. Sb., 70, No. 3, 394–415 (1966).
S. N. Kruzhkov and N. S. Petrosyan, “Asymptotic behavior of solutions of the Cauchy problem for nonlinear partial differential equations of first order,” Usp. Mat. Nauk, 42, 3–40 (1987).
A. V. Kryazhimskii, “On the theory of positional differential games of approach and evasion,” Dokl. Akad. Nauk SSSR, 239, No. 4, 779–782 (1987).
A. V. Kryazhimskii and Yu. S. Osipov, “On positional modelling of controls in dynamical systems,” Izv. Akad. Nauk SSSR, Ser. Tekhn. Kibern., No. 2, 51–60 (1983).
A. V. Kryazhimskii and Yu. S. Osipov, “On differential-evolutionary games,” Proc. Steklov Inst. Math., 211, 234–261 (1995).
A. V. Kryazhimskii and Yu. S. Osipov, “Extremal problems with separable graphs,” Kibern. Syst. Anal., 2, 32–55 (2002).
N. V. Krylov, Controlled Diffusion Processes [in Russian], Nauka, Moscow (1977).
S. I. Kumkov and V. S. Patsko, “Control of informational sets and pursuit problem,” in: Ann. Int. Soc. Dynam. Games, 3, Birkhäuser, Boston (1995), pp. 191–206.
S. S. Kumkov and V. S. Patsko, “Maximal stable bridges to the Pontryagin control example,” Vestn. Udmurt. State Univ., No. 1, 92–103 (2000).
S. I. Kumkov, V. S. Patsko, S. G. Pyatko, and A. A. Fedotov, “Construction of the stable bridge in a problem of aircraft guiding under wind disturbances,” in: Proc. X Int. Symp. “Dynamical Games and Applications,” July 8–11, 2002, St. Petersburg State University, 2, St. Petersburg (2002), pp. 474–480.
A. B. Kurzhanskii, Control and Observation under Conditions of Uncertainty [in Russian], Nauka, Moscow (1979).
A. B. Kurzhanskii and T. F. Filippova, “On the method of singular perturbations to differential inclusions,” Dokl. Akad. Nauk SSSR, 321, No. 3, 454–459 (1991).
A. B. Kurzhanski and T. F. Filippova, “On the theory of trajectory tubes — a mathematical formalism for uncertain dynamics, viability and control,” in: Advances in Nonlinear Dynamics and Control: a Report from Russia, Progr. Systems and Control Theory, 17, Birkhäuser, Boston (1993), pp. 122–188.
A. B. Kurzhanskii and O. I. Nikonov, “Evolutionary equations for bundles of trajectories of designed control systems,” Dokl. Ross. Akad. Nauk, 333, No. 5 (1993).
A. B. Kurzhanskii and I. F. Sivergina, “The dynamical programming method to inverse estimate problems for distributed systems,” Dokl. Ross. Akad. Nauk, 369, No. 2, 161–166 (1998).
A. B. Kurzhanski and P. Varaiya, “On reachability under uncertainty,” SIAM J. Control. Optimiz., 41, No. 1, 181–216 (2002).
N. N. Kuznetsov and B. L. Rozhdestvenskii, Construction of a generalized solution of the Cauchy problem for a quasi-linear equation,” Usp. Mat. Nauk, 14, No. 2, 211–215 (1959).
O. A. Ladyshenskaya and N. N. Uraltseva, Linear and Quasi-linear Partial Differential Equations of Elliptic Type [in Russian], Nauka, Moscow (1964).
O. A. Ladyshenskaya, V. A. Solonnikov, and N. N. Uraltseva, Linear and Quasi-linear Partial Differential Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).
A. S. Lakhtin and A. I. Subbotin, “Multivalued solutions of first-order partial differential equations,” Mat. Sb., 189, No. 6, 33–58 (1998).
A. S. Lakhtin and A. I. Subbotin, “Minimax and viscosity solutions to discontinuous first-order partial differential equations,” Dokl. Ross. Akad. Nauk, 359, No. 4, 452–455 (1998).
P. Lax, “Hyperbolic systems of conservations laws, II,” Commun. Pure Appl. Math., 10, 537–566 (1957).
Yu. S. Ledyaev and E. F. Mishchenko, “Extremal problems in the theory of differential games,” Proc. Steklov Inst. Math., 85, 147–170 (1988).
G. Leitmann, Introduction to the Theory of Optimal Control [in Russian], Nauka, Moscow (1968).
G. Leitmann, “One approach to the control of uncertain dynamical systems,” Appl. Math. Comput., 70, 261–272 (1995).
K. Leichtveis, Convex Sets [in Russian] Nauka, Moscow (1985).
E. F. Lelikova, “On asymptotics of the fundamental solution to a parabolic partial differential equation of higher order,” Dokl. Ross. Akad. Nauk, 341, No. 5, 532–537 (1995).
P. L. Lions, Generalized Solutions of Hamilton—Jacobi Equations, Pitman Research Notes, Math. Ser., 69, Pitman, Boston (1982).
P. L. Lions, “Optimal control of diffusion processes and Hamilton—Jacobi—Bellman equations, 2,” Commun. Part. Differ. Equations, 8, No. 11, 1229–1276 (1983).
P. L. Lions and P. E. Souganidis, “Differential games, optimal control and directional derivatives of viscosity solutions of Bellman’s and Isaacs’s equations,” SIAM J. Control Optimiz., 23, No. 4, 566–583 (1985).
R. Sh. Lipzer and A. N. Shiryaev, Statistics of Random Processes [in Russian] Nauka, Moscow (1974).
M. I. Loginov, O. N. Sobolev, and G. S. Shelement’ev, Introduction to Statistical Analysis [in Russian], Ural State Univ. Ekaterinburg (1999).
N. Yu. Lukoyanov, “Minimax solutions of the Hamilton—Jacobi equations with a heredity,” Dokl. Ross. Akad. Nauk, 371, No. 2, 163–166 (2000).
V. I. Maksimov, “The principle of extrimal shift in a problem of solving operator equations,” Proc. Steklov Inst. Mat., Suppl. Issue 1, S163–S171 (2000).
O. A. Malafeyev, “Perfect equilibrium in noncooperative differential games,” in: Proc. X Int. Symp. “Dynamical Games and Applications,” St.Petersburg State University, 2002, 2, St.Petersburg (2002), pp. 492–494.
V. V. Malyshev and A. I. Kibzun, Analysis and Synthesis of High-Precision Controls of Aircraft [in Russian], Moscow (1987).
V. P. Maslov and S. N. Samborskii, “Existence and uniqueness of solutions to the steady-state Hamilton—Jacobi—Bellman equations. A new approach,” Dokl. Ross. Akad. Nauk, 324, No. 6, 1143–1148 (1992).
A. A. Melikyan, “Singular characteristics first-order partial differential equations,” Dokl. Ross. Akad. Nauk, 382, No. 2, 203–217 (1996).
A. A. Melikyan, Generalized Characteristics of First Order Partial Differential Equations: Applications in Optimal Control and Differential Games, Birkkäuser Boston (1998).
A. A. Melikyan, “Equations of propagations of weak gaps of a solution to a variational problem,” Proc. Steklov Inst. Mat., Suppl. Issue 2, S446–S459 (2000).
L. I. Minchenko and O. F. Borisenko, Differential Properties of Marginal Functions and Applications to Optimization Problems [in Russian], Minsk (1992).
S. Mirică, “Extending Cauchy’s method of characteristics for Hamilton—Jacobi equations,” Stud. Cerc. Mat., 37, No. 6, 555–565 (1985).
E. F. Mishchenko, “Pursuit-evasion problems in the theory of differential games,” Izv. Akad. Nauk SSSR. Ser. Tekhn. Kibern., No. 5, 3–9 (1971).
E. F. Mishchenko and L. S. Pontryagin, “Periodic solutions of near discontinuous systems of ordinary differential equation,” Dokl. Alad. Nauk SSSR, 102, No. 5, 889–891 (1955).
E. F. Mishchenko and N. Kh. Rosov, Differential Equations with a Small Parameter and Relaxational Oscillations [in Russian], Nauka, Moscow (1975).
N. N. Moiseev, Asymptotic Methods of Nonlinear Mechanics [in Russian], Nauka, Moscow (1971).
B. Sh. Mordukhovich, Methods of Appoximations to Optimization and Control Problems [in Russian], Nauka, Moscow (1988).
A. A. Neznakhin and V. N. Ushakov, “A grid method to approximations of the kernel of viability of a differential inclusion,” Zh. Vychisl. Mat. Mat. Fiz., 41, No. 6, 895–908 (2001).
M. S. Nikol’skii, “On the Pontryagin alternated integral,” Mat. Sb., 116, No. 1, 136–144 (1981).
M. S. Nikol’skii and M. Aboubacar, “Some estimates of the reachable set for the controlled Van der Pol equation,” Proc. Steklov Inst. Math., Suppl. Issue 1, S172–S181 (2000).
O. A. Oleinik, “Discontinuous solutions to nonlinear differental equations,” Usp. Mat. Nauk, 12, No. 3, 3–73 (1957).
O. A. Oleinik, “On construction of a generalized solution to the Cauchy problem for a quasi-linear first-order partial differential equation involving vanishing viscosity,” Usp. Math. Nauk, 14, No. 2, 159–164 (1959).
O. A. Oleinik and S. N. Kruzhkov, “Quasi-linear parabolic partial differential equations of second order with several independent variables,” Usp. Mat. Nauk, 16, No. 5, 115–155 (1961).
R. O’Malley, Introduction to Singular Perturbations, Academic Press, New York (1974).
R. Ortega, A. J. van der Schaft, and B. M. Maschke, Stabilization of Port-Controlled Hamiltonian Systems via Energy Balancing. in Stability and Stabilization of Nonlinear Systems. vol. 246. LNCIS. New York: Springer Verlag. 1999.
Yu. S. Osipov, “On the theory of differential games in systems with distributed parameters,” Dokl. Akad. Nauk SSSR, 223, No. 6, 1314–1317 (1975).
Yu. S. Osipov, “Positional control in parabolic systems,” Prikl. Mat. Mekh., 41, No. 2 (1977).
A. I. Panasyuk and V. I. Panasyuk, Asymptotic Magistral Optimization in Control Systems [in Russian], Nauka, Moscow (1986).
T. Parthasarathy and T. Raghavan, Some Topics in Two-Person Games, Modern Analytic and Computational Methods in Science and Mathematics, 22, Amer. Elsevier, New York (1971).
A. G. Pashkov and S. D. Terekhov, “Differential game of approach with two pursuers and one evader,” J. Optimiz. Theory Appl., 55, No. 2, 303–311 (1987).
V. S. Patsko and V. L. Turova, “Level sets of the value function in differential games with homicidal chauffeur dynamics,” Int. Game Theory Rev., 3, No. 1, 67–112 (2001).
A. A. Pervozvanskii and V. G. Gaitsgory, Decomposition, Aggregation, and Approximation Optimization [in Russian] Nauka, Moscow (1979).
A. A. Pervozvanski and V. G. Gaitsgory, Theory of Suboptimal Solutions, Kluwer Academic, Dordrecht (1988).
L. A. Petrosyan, Pursuit Differential Games [in Russian], Leningrad State Univ., Leningrad (1977).
L. A. Petrosyan and V. V. Zakharov, Mathematical Models in Ecology [in Russian], St.Petersburg State Univ., St.-Petersburg (1997).
N. N. Petrov, “On the existence of the value in a pursuit game,” Dokl. Akad. Nauk SSSR, 190, No. 6, 621–624 (1970).
N. N. Petrov, Game Theory [in Russian], Udmurt. State Univ., Izhevsk (1997).
I. G. Petrovskii, Lectures on the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1964).
E. S. Polovinkin, Elements of the Theory of Mulivalued Mappings [in Russian], Moscow (1982).
B. T. Polyak, An Introduction to Optimization [in Russian], Nauka, Moscow (1983).
L. S. Pontryagin, Ordinary Differential Equations [in Russian], Nauka, Moscow (1965).
L. S. Pontryagin, “On linear differential games, 1, 2,” Dokl. Akad. Nauk SSSR, 174, No. 6, 1278–1280; 175, No. 4, 764–766 (1967).
L. S. Pontryagin, “Mathematical theory of optimal processes and differential games,” Proc. Steklov Inst. Mat., 169, 119–157 (1985).
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1961).
B. N. Pshenichnyi, “The structure of differential games,” Dokl. Akad. Nauk SSSR, 184, No. 2, 285–287 (1969).
B. N. Pshenichnyi, Convex Analysis and Extremal Problems [in Russian], Nauka, Moscow (1980).
B. N. Pshenichnyi and V. V. Ostapenko, Differential Games [in Russian], Naukova Dumka, Kiev (1992).
R. T. Rockafellar, Convex Analysis [Russian translation], Mir, Moscow (1973).
R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer-Verlag, New York (1998).
R. T. Rockafellar and P. R. Wolenski, Convexity and duality in Hamilton—Jacobi theory, Report of Int. Inst. Appl. Syst. Anal. IR-98-057, Laxenburg (1998).
J. D. L. Rowland and R. B. Vinter, “Constructions of optimal feedback controls,” Systems Control Lett., 16, 357–367 (1991).
E. Roxin, “The axiomatic approach in differential games,” J. Optimiz. Theory Appl., 3, 153–163 (1969).
B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasi-Linear Equations and Applications to Gas Dynamics [in Russian], Nauka, Moscow (1978).
L. I. Rozonoer, “The Pontryagin maximum principle in the theory of optimal systems,” Automat. Telemekh., 20, No. 10–12 (1959).
L. Schwartz, Analysis [Russian translation], Mir, Moscow (1972).
A. N. Shiryaev, Probability [in Russian], Nauka, Moscow (1980).
A. F. Shorikov, Minimax Estimates and Control of Discrete Dynamical Systems [in Russian], Ural State Univ., Ekaterinburg (1997).
A. F. Sidorov, “On optimal unshocked compression of gas flows,” Dokl. Akad. Nauk SSSR, 313, No. 2, 283–287 (1990).
A. F. Sidorov, V. L. Gasilov, and A. P. Kukushkin, “Development of high-efficiency algorithms and software on the basis of parallel technologies,” in: Algorithms and Software of Parallel Calculations [in Russian], Ekaterinburg (1995), pp. 3–20.
E. V. Sidorova and N. N. Subbotina, “An algorithm for calculating the value function in a linear differential game with a convex cost function,” in: Positional Control with Guaranteed Result [in Russian], Sverdlovsk (1986), pp. 62–74.
S. L. Sobolev, On Some Applications of Functional Analysis to Mathematical Physics [in Russian] Nauka, Moscow (1988).
G. Sonnevend, On constructing feedback control in differential games by use of central trajectories, Report Schwerpunktprogr. DFG Anwendungsbezogene Optimierung und Steuerung, No. 385, Würzburg (1992).
P. E. Souganidis, “Max-min representations and product formulas for the viscosity solutions of Hamilton—Jacobi equations with applications to differential games,” Nonlinear Analysis. Theory, Meth. Appl., 9, No. 3, 217–257 (1985).
V. V. Stepanov, A Course in Differential Equations [in Russian], Fizmatgiz, Moscow (1959).
J. Stoer and C. Witzgall, Convexity and Optimization in Finite Dimensions, I, Springer-Verlag, New York (1970).
A. I. Subbotin, “On a generalization of the main equation to the theory of differential games,” Dokl. Akad. Nauk SSSR, 254, No. 2, 293–297 (1980).
A. I. Subbotin, Minimax Inequalities and the Hamilton—Jacobi Equations [in Russian], Nauka, Moscow (1991).
A. I. Subbotin, “Continuous and discontinuous solutions to boundary problems for first-order partial differential equations,” Dokl. Ross. Akad. Nauk, 323, No. 1, 30–34 (1992).
A. I. Subbotin, Generalized Solutions of First-Order Partial Differential Equations: The Dynamical Optimization Perspective, Birkhäuser, Boston (1995).
A. I. Subbotin, “Minimax solutions of first-order partial differential equations,” Usp. Mat. Nauk, 51, No. 2, 105–138 (1996).
A. I. Subbotin and A. G. Chentsov, Optimization of the Guarantee in Control Problems [in Russian], Nauka, Moscow (1981).
A. I. Subbotin and A. G. Chentsov, “An iteration procedure for constructing minimax and viscosity solutions to the Hamilton—Jacobi equations,” Dokl. Ross. Akad. Nauk, 348, No. 3, 45–48 (1996).
A. I. Subbotin and L. G. Shagalova, “A piecewise linear solution of the Cauchy problem for the Hamilton—Jacobi equation,” Dokl. Akad. Nauk, 325, No. 5, 932–936 (1992).
A. I. Subbotin and N. N. Subbotina, “Necessary and sufficient conditions for a piecewise smooth value of a differential game,” Dokl. Akad. Nauk SSSR, 243, No. 4, 829–865 (1978).
A. I. Subbotin and N. N. Subbotina, “Necessary and sufficient conditions for nonsmooth value of a differential game,” in: Problems of Dynamical Control [in Russian], Sverdlovsk (1979).
A. I. Subbotin and N. N. Subbotina, “The optimal result function in a control problem,” Dokl. Akad. Nauk SSSR, 266, No. 2, 294–299 (1982).
A. I. Subbotin and N. N. Subbotina, “Properties of a differential game potential,” Prikl. Mat. Mekh., 46, No. 2, 204–211 (1982).
A. I. Subbotin and N. N. Subbotina, “On justification of the method of dynamical programming to an optimal control problem,” Izv. Akad. Nauk SSSR, Ser. Tekhn. Kibern., 2, 24–32 (1983).
A. I. Subbotin and N. N. Subbotina, “Differentiability properties of the value finction in a differential game with an integral-terminal payoff,” Probl. Contr. Inform. Theory, 12, No. 3, 153–166 (1983).
A. I. Subbotin and N. N. Subbotina, “Piecewise smooth solutions of first order partial differential equations,” Dokl. Ross. Akad. Nauk, 333, No. 6, 705–707 (1993).
A. I. Subbotin, A. M. Tarasyev, and V. N. Ushakov, “Generalized characteristics of the Hamilton—Jacobi equations,” Izv. Akad. Nauk, Ser. Tekhn. Kibern., No. 1, 190–197 (1993).
A. I. Subbotin, A. M. Tarasyev, and V. N. Ushakov, “Generalized characteristics of Hamilton—Jacobi equations,” J. Comput. Systems Sci. Int., 32, No. 2, 157–163 (1994).
N. N. Subbotina, “Universal optimal strategies in positional differential games,” Diff. Equations, 19, No. 11, 1377–1382 (1983).
N. N. Subbotina, “Some sufficient conditions for existence of universal strategies,” in: Investigations of Minimax Control Problems [in Russian], Sverdlovsk (1985), pp. 72–81.
N. N. Subbotina, Infinitesimal properties of the value function of a diffusion differential game [in Russian], Inst. Math. Mech., Sverdlovsk, 1985, deposited at the All-Union Institute for Scientific and Technical Information, Moscow (1985), No. 7690-D19.11.85.
N. N. Subbotina, “Necessary and sufficient conditions for optimality of controls and trajectories,” in: Synthesis of Optimal Control in Game Systems [in Russian], Sverdlovsk (1986).
N. N. Subbotina, “Generalized stochastic derivatives to functions of several variables differential in a part of the variables,” in: Control with Guaranteed Result [in Russian], Sverdlovsk (1987), pp. 77–85.
N. N. Subbotina, Necessary and sufficient optimality conditions in terms of the maximum principle and superdifferetial of the value function [in Russian], Inst. Math. Mech., Sverdlovsk, 1988, deposited at the All-Union Institute for Scientific and Technical Information, Moscow (1988), No. 2898-B88.
N. N. Subbotina, “The maximum principle and the superdifferential of the value function,” Probl. Control Inform. Theory, 18, No. 3, 151–160 (1989).
N. N. Subbotina, “The Cauchy method of characteristics and generalized solutions of the Hamilton—Jacobi—Bellman equations,” Dokl. Akad. Nauk SSSR, 320, No. 3, 556–561 (1991).
N. N. Subbotina, Constructing a generalized solution of the Hamilton—Jacobi—Bellman equation by the Cauchy method of characteristics [in Russian], Inst. Math. Mech., Sverdlovsk (1991), deposited at the All-Union Institute for Scientific and Technical Information, Moscow (1991), No. 2571-B91.
N. N. Subbotina, “Unified optimality conditions to control problems,” in: Proc. Inst. Math. Mech., Ural Branch Russ. Acad. Sci. [in Russian], 1, Ekaterinburg (1992), pp. 147–160.
N. N. Subbotina, Necessary and sufficient optimality conditions in terms of characteristics of Hamilton—Jacobi—Bellman equation, Report Schwerpunktprogr. DFG Anwendungsbezogene Optimierung und Steuerung, No. 393, Würzburg (1992).
N. N. Subbotina, “Asymptotic properties of minimax solutions to the Hamilton—Jacobi equations in differential games with fast and slow motions,” Prikl. Mat. Mekh., 60, No. 6, pp. 883–890 (1996).
N. N. Subbotina, “Asymptotics for singularly perturbed differential games,” in: Proc. IFAC Conference “Nonsmooth and Discontinuous Problems of Control and Optimization, June 17–20, 1998, Chelyabinsk, Russia, Elsevier Science, Oxford (1998), pp. 43–52.
N. N. Subbotina, “Asymptotics of the singularly perturbed Hamilton—Jacobi equations,” Prikl. Mat. Mekh., 63, No. 2, 220–230 (1999).
N. N. Subbotina, “Singular approximations of the minimax and viscosity solutions to the Hamilton—Jacobi equations,” Proc. Steklov Institute of Mathematics, Suppl. Issue 1, S210–S227 (2000).
N. N. Subbotina, “On structure of optimal feedbacks to control problems,” in: Proc. 11 IFAC Int. Workshop “Control Applications of Optimization,” July 3–6, 2000, St.Petersburg State University, St.-Petersburg (2000), pp. 254–255.
N. N. Subbotina, “On structure of optimal feedbacks to control problems,” in: Proc. 11 IFAC Int. Workshop “Control Applications of Optimization,” 1, St. Petersburg State Univ. (2000), pp. 339–344.
N. N. Subbotina, “Generalized Cauchy characteristics to singularly perturbed Hamilton—Jacobi equations,” in: Proc. Int. Conf. “Differential Equations and Dynamical Systems,” August 21–26, 2000, Suzdal, Steklov Inst. Math., Vladimir State Univ. (2000), pp. 93–94.
N. N. Subbotina, “Asymptotics for singularly perturbed differential games,” Game Theory Appl., 7, 175–196 (2001).
N. N. Subbotina, “Near optimal feedbacks to mechanical systems with fast and slow motions,” in: Proc. VIII Symp. on Theoretical and Applied Mechanics, August 23–29, 2001, Perm, Ekaterinburg (2001), p. 546.
N. N. Subbotina, Optimality conditions for feedbacks to control problems, Inst. Math. Mech., Ekaterinburg (2002), deposited at the All-Union Institute for Scientific and Technical Information, Moscow (2002), No. 1212-B2002.
N. N. Subbotina, “Sufficient optimality conditions for feedbacks to control problems,” in: Proc. X Int. Symp. “Dynamical Games and Applications,” July 8–11, 2002, St. Petersburg State University, 2, St. Petersburg (2002), pp. 829–834.
N. N. Subbotina, “The method of dynamical programming for a class of local Lipschitz functions,” Dokl. Ross. Akad. Nauk, 389, No. 2, 169–172 (2003).
N. N. Subbotina and A. G. Chentsov, “On the existence of the Bellman function in a linear differential game,” in: Proc. Inst. Math. Mech. [in Russian], No. 26, Sverdlovsk (1979), pp. 80–86.
N. N. Subbotina, A. I. Subbotin, and V. E. Tretyakov, “Stochastic and deterministic control. Differential inclusions,” Probl. Control Inform. Theory, 14, No. 6, P1–P15 (1985).
N. N. Subbotina, A. I. Subbotin, and V. E. Tretyakov, “Stochastic and deterministic control. Differential inequalities,” Lect. Notes Control Inform., 81, 728–737 (1987).
N. N. Subbotina and V. N. Ushakov, “On characteristic differential inclusions to unified differential games,” in: Proc. IX Int. Symp. “Dynamical Games and Applications,” December 18–21, 2000, Adelaida, Australia, Univ. of South Australia, Adelaide (2000), pp. 464–467,.
A. M. Tarasyev, “Approximation schemes for constructing minimax solutions to the Hamilton—Jacobi equations,” Prikl. Mat. Mekh., 58, No. 2, 22–36 (1994).
A. M. Tarasyev, A. A. Uspenskii, and V. N. Ushakov, “Approximation schemes and finite difference operators for constructing generalized solutions of the Hamilton—Jacobi equations,” Izv. Ross. Akad. Nauk, Ser. Tekhn. Kibern., No. 3, 173–185 (1994).
A. N. Tikhonov, “Systems of differential equations with a small parameter at derivatives,” Mat. Sb., 31, No. 3, 575–586 (1952).
A. N. Tikhonov and A. A. Samarskii, “On discontinuous solutions of a quasi-linear first-order partial differential equation,” Dokl. Akad. Nauk SSSR, 99, No. 1, 27–30 (1954).
E. L. Tonkov, “Some questions in control of periodic motions,” in: Dynamics of Control Systems [in Russian], Nauka, Novosibirsk (1979).
V. E. Tretyakov, “On the theory of stochastic differential games,” Dokl. Akad. Nauk SSSR, 269, No. 3, 1049–1053 (1983).
V. E. Tretyakov, I. V. Tzelishcheva, and G. I. Shishkin, “Optimal control in systems with incomplete and uncertain information,” in: Proc. Inst. Math. Mech., 2, Ekaterinburg (1992), pp. 176–187.
V. A. Troitskii, Optimal Oscillation Processes of Mechanical Systems [in Russian], Leningrad (1976).
V. I. Ukhobotov, “Synthesis of guaranteed controls on the basis of an approximation scheme,” Proc. Steklov Math. Inst., Suppl. Issue 1, S254–S260 (2000).
V. N. Ushakov, “On a problem of constructing stable bridges to a pursuit-evasion differential game,” Izv. Akad. Nauk SSSR, Ser. Tekhn. Kibern., No. 4, 29–36 (1980).
V. N. Ushakov, “On a question of stability in differential games,” in: Positional Control with Guaranteed Result [in Russian], Sverdlovsk (1988), pp. 101–109.
V. N. Ushakov, “Constructions of solutions in differential games of pursuit-evasion. Differential inclusions and optimal control,” Lect. Notes Nonlin. Anal., 2, 269–281 (1998).
V. N. Ushakov and A. P. Khripunov, “On constructing approximation solutions to game-theoretic control problems,” Prikl. Mat. Mekh. 61, No. 3, 413–421 (1997).
V. I. Utkin, Sliding Behaviors and Applications to Systems with Changing Structure [in Russian], Nauka, Moscow (1974).
V. A. Vakhrushev and V. N. Ushakov, “On simulations of control procedures with a guide,” Prikl. Mat. Mekh., 66, No. 2, 228–238 (2002).
P. Varaiya, “On the existence of solutions to a differential game,” SIAM J. Control Optimiz., 5, No. 1, 153–162 (1967).
A. B. Vasil’eva and V. F. Butuzov, Asymptotic Expansions of Singularly Perturbed Equations [in Russian] Nauka, Moscow (1973).
V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1976).
V. Veliov, “A generalization of the Tikhonov theorem for singularly perturbed differential inclusions,” J. Dynam. Control Systems, 3, 291–319 (1997).
V. Veliov, “Stability-like properties of differential inclusions,” Set-Valued Anal., 5, No. 1, 73–88 (1997).
V. A. Vyazgin, “On justification of sufficient conditions with the help of the Weierstrass and Hamilton—Jacobi—Bellman methods,” Automat. Telemech., 4, 31–37 (1984).
J. Warga, Optimal Control of Differential and Functional Equations [Russian translation], Nauka, Moscow (1977).
L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory [Russian translation], Mir, Moscow (1974).
S. T. Zavalishchin and A. N. Sesekin, Impulse Processes. Models and Applications [in Russian], Nauka, Moscow (1991).
M. I. Zelikin, Optimal Control and Variational Calculus [in Russian], Nauka, Moscow (1985).
L. F. Zelikina, “Universal manifolds and theorems on a magistral way to a class of optimal control problems,” Dokl. Akad. Nauk SSSR, 224, No. 1 (1975).
O. A. Zhautykov, V. I. Zhukovskii, and S. Zharkynbaev, Differential Games with Several Persons [in Russian], Nauka, Alma-Ata (1988).
X.-Y. Zhou, “Maximum principle, dynamical programming, and their connection in deterministic controls,” J. Optimiz. Theory Appl., 65, 363–373 (1990).
V. I. Zubov, Lectures on Control Theory [in Russian], Nauka, Moscow (1975).
Author information
Authors and Affiliations
Additional information
Dedicated to the shiny memory of my parents Nikolay Maximovich and Zoya Nikolaevna Barabanovs
__________
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 20, Differential Equations, 2004.
Rights and permissions
About this article
Cite this article
Subbotina, N.N. The method of characteristics for Hamilton—Jacobi equations and applications to dynamical optimization. J Math Sci 135, 2955–3091 (2006). https://doi.org/10.1007/s10958-006-0146-2
Issue Date:
DOI: https://doi.org/10.1007/s10958-006-0146-2