The minimum principle of hybrid optimal control theory

Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes, vol 4. Wiley Interscience, New York Puri A, Varaiya P (1994) Verification of hybrid systems using abstractions. In: International Hybrid Systems Workshop. Springer, pp. 359–369 Alur R, Henzinger TA, Lafferriere G, Pappas GJ (2000) Discrete abstractions of hybrid systems. Proc IEEE 88(7):971–984 Alur R, Dang T, Ivančić F (2003) Progress on reachability analysis of hybrid systems using predicate abstraction. In: Proceedings of the 6th international workshop on hybrid systems: computation and control. HSCC, Prague, Czech Republic, p. 4–19 Clarke E, Fehnker A, Han Z, Krogh B, Ouaknine J, Stursberg O et al (2003) Abstraction and counterexample-guided refinement in model checking of hybrid systems. Int J Found Comput Sci 14(04):583–604 Tiwari A, Khanna G (2002) Series of abstractions for hybrid automata. In: Proceedings of the 5th international workshop on hybrid systems: computation and control. HSCC, Stanford, pp. 465–478 Broucke M (1999) A geometric approach to bisimulation and verification of hybrid systems. In: Hybrid systems: computation and control. Springer, pp. 61–75 Helwa MK, Caines PE (2017) In-block controllability of affine systems on polytopes. IEEE Trans Autom Control 62(6):2950–2957 Corona D, Giua A, Seatzu C (2004) Optimal control of hybrid automata: design of a semiactive suspension. Control Eng Pract 12(10):1305–1318 (Analysis and Design of Hybrid Systems) Goebel R, Sanfelice RG, Teel AR (2012) Hybrid dynamical systems: modeling, stability, and robustness. Princeton University Press, Princeton Liberzon D (2003) Switching in systems and control, vol 190. Birkhauser, Boston Liberzon D, Hespanha JP, Morse AS (1999) Stability of switched systems: a lie-algebraic condition. Syst Control Lett 37(3):117–122 Hespanha JP (2004) Uniform stability of switched linear systems: extensions of LaSalle’s invariance principle. IEEE Trans Autom Control 49(4):470–482 Branicky MS (1998) Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans Autom Control 43(4):475–482 Decarlo RA, Branicky MS, Pettersson S, Lennartson B (2000) Perspectives and results on the stability and stabilizability of hybrid systems. Proc IEEE 88(7):1069–1082 Johansson M, Rantzer A (1998) Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans Autom Control 43(4):555–559 Van der Schaft AJ, Schumacher JM (2000) An introduction to hybrid dynamical systems. Lecture notes in control and information sciences, Vol. 251. Springer, London Bensoussan A, Menaldi JL (1997) Hybrid control and dynamic programming. Dyn Cont Discrete Impuls Syst Ser B Appl Algorithm 3(4):395–442 Dharmatti S, Ramaswamy M (2005) Hybrid control systems and viscosity solutions. SIAM J Control Optim 44(4):1259–1288 Barles G, Dharmatti S, Ramaswamy M (2010) Unbounded viscosity solutions of hybrid control systems. ESAIM Control Optim Calc Var 16(1):176–193 Branicky MS, Borkar VS, Mitter SK (1998) A unified framework for hybrid control: model and optimal control theory. IEEE Trans Autom Control 43(1):31–45 Shaikh MS, Caines PE (2009) A verification theorem for hybrid optimal control problem. In: Proceedings of the IEEE 13th international multitopic conference, INMIC Caines PE, Egerstedt M, Malhamé R, Schöllig A (2007) A hybrid Bellman equation for bimodal systems. In: Proceedings of the 10th international conference on hybrid systems: computation and control, HSCC, vol. 4416, p. 656–659. LNCS Schöllig A, Caines PE, Egerstedt M, Malhamé R (2007) A hybrid Bellman equation for systems with regional dynamics. In: Proceedings of the 46th IEEE conference on decision and control, CDC. pp. 3393–3398 Da Silva JE, De Sousa JB, Pereira FL (2012) Dynamic programming based feedback control for systems with switching costs. In: Proceedings of the IEEE international conference on control applications, CCA, pp. 634–639 Hedlund S, Rantzer A (2002) Convex dynamic programming for hybrid systems. IEEE Trans Autom Control 47(9):1536–1540 Clarke FH, Vinter RB (1989) Applications of optimal multiprocesses. SIAM J Control Optim 27(5):1048–1071 Clarke FH, Vinter RB (1989) Optimal multiprocesses. SIAM J Control Optim 27(5):1072–1091 Sussmann HJ (1999) A nonsmooth hybrid maximum principle. In: Aeyels D, Lamnabhi-Lagarrigue F, van der Schaft A (eds) Stability and stabilization of nonlinear systems. London, Springer, pp 325–354 Sussmann HJ (1999) Maximum principle for hybrid optimal control problems. In: Proceedings of the 38th IEEE conference on decision and control, CDC. pp. 425–430 Caines PE, Clarke FH, Liu X, Vinter RB (2006) A maximum principle for hybrid optimal control problems with pathwise state constraints. In: Proceedings of the 45th IEEE conference on decision and control, pp. 4821–4825 Shaikh MS, Caines PE (2007) On the hybrid optimal control problem: theory and algorithms. IEEE Trans Autom Control 52(9):1587–1603 Garavello M, Piccoli B (2005) Hybrid necessary principle. SIAM J Control Optim 43(5):1867–1887 Taringoo F, Caines PE (2013) On the optimal control of impulsive hybrid systems on Riemannian manifolds. SIAM J Control Optim 51(4):3127–3153 Pakniyat A, Caines PE (2017) On the relation between the minimum principle and dynamic programming for classical and hybrid control systems. IEEE Trans Autom Control 62(9):4347–4362 Jafarpour S, Lewis AD (2016) Locally convex topologies and control theory. Math Control Signals Syst 28(4):29 Shaikh MS, Caines PE (2005) Optimality zone algorithms for hybrid systems computation and control: from exponential to linear complexity. In: Proceedings of the 44th IEEE conference on decision and control, and the european control conference, CDC-ECC ’05. vol, 2005, pp. 1403–1408 Taringoo F, Caines PE (2011) Gradient geodesic and Newton geodesic HMP algorithms for the optimization of hybrid systems. Annu Rev Control 35(2):187–198 Axelsson H, Wardi Y, Egerstedt M, Verriest E (2008) Gradient descent approach to optimal mode scheduling in hybrid dynamical systems. J Optim Theory Appl 136(2):167–186 Boccadoro M, Wardi Y, Egerstedt M, Verriest E (2005) Optimal control of switching surfaces in hybrid dynamical systems. Discrete Event Dyn Syst 15(4):433–448 Gonzalez H, Vasudevan R, Kamgarpour M, Sastry SS, Bajcsy R, Tomlin CJ (2010) A descent algorithm for the optimal control of constrained nonlinear switched dynamical systems. In: Proceedings of the 13th ACM international conference on Hybrid systems: computation and control, ACM, pp. 51–60 Zhao P, Mohan S, Vasudevan R (2019) Optimal control of polynomial hybrid systems via convex relaxations. IEEE Trans Autom Control 65(5):2062–2077 Zhu F, Antsaklis PJ (2015) Optimal control of hybrid switched systems: a brief survey. Discret Event Dyn Syst 25(3):345–364 Passenberg B, Leibold M, Stursberg O, Buss M (2011) The minimum principle for time-varying hybrid systems with state switching and jumps. In: Proceedings of the 50th IEEE conference on decision and control and european control conference, CDC-ECC. pp. 6723–6729 Cowlagi RV (2017) Hierarchical trajectory optimization for a class of hybrid dynamical systems. Automatica 77:112–119 Mamakoukas G, MacIver MA, Murphey TD (2018) Feedback synthesis for underactuated systems using sequential second-order needle variations. Int J Robot Res 37(13–14):1826–1853 Riedinger P, Kratz F (2003) An optimal control approach for hybrid systems. Eur J Control 9(5):449–458 Xu X, Antsaklis PJ (2004) Optimal control of switched systems based on parameterization of the switching instants. IEEE Trans Autom Control 49(1):2–16 Azhmyakov V, Boltyanski VG, Poznyak A (2008) Optimal control of impulsive hybrid systems. Nonlinear Anal Hybrid Syst 2(4):1089–1097 Dmitruk AV, Kaganovich AM (2008) The hybrid maximum principle is a consequence of Pontryagin maximum principle. Syst Control Lett 57(11):964–970 Dmitruk AV, Kaganovich AM (2011) Maximum principle for optimal control problems with intermediate constraints. Comput Math Model 22(2):180–215 Dmitruk AV, Kaganovich AM (2011) Quadratic order conditions for an extended weak minimum in optimal control problems with intermediate and mixed constraints. Discr Contin Dyn Syst 29:523–545 Pakniyat A, Caines PE (2014) On the relation between the minimum principle and dynamic programming for hybrid systems. In: Proceedings of the 53rd IEEE conference on decision and control, CDC. pp. 19–24 Pakniyat A, Caines PE (2013) The hybrid minimum principle in the presence of switching costs. In: Proceedings of the 52nd IEEE conference on decision and control, CDC. pp. 3831–3836 Westervelt ER, Chevallereau C, Choi JH, Morris B, Grizzle JW (2007) Feedback control of dynamic bipedal robot locomotion. CRC Press, Boca Raton Pakniyat A, Caines PE (2017) Hybrid optimal control of an electric vehicle with a dual-planetary transmission. Nonlinear Anal Hybrid Syst 25:263–282 Pakniyat A, Caines PE (2015) Time optimal hybrid minimum principle and the gear changing problem for electric vehicles. In: Proceedings of the 5th IFAC conference on analysis and design of hybrid systems. Atlanta, pp. 187–192 Pakniyat A, Caines PE (2014) On the minimum principle and dynamic programming for hybrid systems. In: Proceedings of the 19th international federation of automatic control world congress, IFAC, p. 9629–9634 Pakniyat A, Caines PE (2015) On the minimum principle and dynamic programming for hybrid systems with low dimensional switching manifolds. In: Proceedings of the 54th IEEE conference on decision and control, Japan. pp. 2567–2573 Pakniyat A, Caines PE (2015) On the relation between the hybrid minimum principle and hybrid dynamic programming: a linear quadratic example. In: Proceedings of the 5th IFAC conference on analysis and design of hybrid systems. pp. 169–174 Taringoo F, Caines PE (2010) Gradient-geodesic HMP algorithms for the optimization of hybrid systems based on the geometry of switching manifolds. In: Proceedings of the 49th IEEE conference on decision and control, CDC. pp. 1534–1539 Pakniyat A, Caines PE (2016) On the stochastic minimum principle for hybrid systems. In: Proceedings of the 55th IEEE conference on decision and control. pp. 1139–1144 Caines PE (2017) Lecture notes on nonlinear and hybrid control systems: dynamics, stabilization and optimal control. Department of Electrical and Computer Engineering (ECE), McGill University Agrachev AA, Sachkov Y (2013) Control theory from the geometric viewpoint, vol 87. Springer Science & Business Media, Berlin Rudin W (1987) Real and complex analysis. McGraw-Hill, New York Sontag ED (1998) Mathematical control theory: deterministic finite dimensional systems. texts in applied mathematics. Springer, New York Pakniyat A, Caines PE (2020) On the hybrid minimum principle: the Hamiltonian and adjoint boundary conditions. IEEE Trans Autom Control 66(3):1246–1253