Computational methods for optimal control

Nonlinear dynamical systems abound in the natural and technological world. These systems can exhibit highly complex behaviour that can be extremely difficult to characterize and control in worst-case scenarios. Computational intractability is the major obstacle that prevents the practical application of standard worst-case analysis and optimal control tools to this end. This project is concerned with the development of new theory and computationally efficient methods for understanding and controlling the worst-case behaviour of nonlinear dynamical systems. Advances in idempotent algebra and dynamic programming are fundamental to this development. 

Recent sponsorship: US Dept. of Air Force (AFOSR).

Selected preprints and new publications

• 1. P.M. Dower, H. Kaise, W.M. McEneaney, T. Wang, and R. Zhao. Solution existence and uniqueness for degenerate SDEs with application to Schrodinger equation representations. To appear, Communications in Information and Systems [20 pages], 2021.
• 2. V. Basco, P.M. Dower, W.M. McEneaney, and I. Yegorov. Exploiting characteristics in stationary action problems. To appear, Applied Mathematics and Optimization [31 pages], 2021.
• 3. I. Yegorov, P.M. Dower, and L. Grune. Synthesis of control Lyapunov functions and stabilizing feedback strategies using exit-time optimal control: Part I. Optimal Control Applications and Methods [25 pages, DOI:10.1002/oca.2732], 2021.
• 4. I. Yegorov, P.M. Dower, and L. Grune. Synthesis of control Lyapunov functions and stabilizing feedback strategies using exit-time optimal control: Part II. To appear, Optimal Control Applications and Methods [30 pages, DOI:10.1002/oca.2733], 2021.
• 5. J. Kennedy, P.M. Dower, and A. Chapman. Convergence rates of coverage controllers. In review [6 pages], 2021.
• 6. J. Darbon, P.M. Dower, and T. Meng. Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs. Prepeint [41 pages, arXiv:2105.03336], 2021.
• 7. P.M. Dower, W.M. McEneaney, and M. Cantoni. Game representations for state constrained continuous time linear regulator problems. Preprint [34 pages, arXiv:1904.05552], 2021.

Recent publications

• 1. P.M. Dower and W.M. McEneaney. Verifying fundamental solution groups for lossless wave equations via stationary action and optimal control. Applied Mathematics and Optimization [DOI:10.1007/s00245-020-09700-4], 2020.
• 2. I. Yegorov and P.M. Dower. Characteristics in Optimal Control Computation, pages 1–10. Encyclopedia of Systems & Control. Springer [DOI:10.1007/978-1-4471-5102-9_100056-1], 2020.
• 3. P.M. Dower and W.M. McEneaney. Verification of stationary action trajectories via optimal control. Proc. American Control Conference (Denver CO), 2020.
• 4. P.M. Dower and W.M. McEneaney. A min-plus fundamental solution semigroup for a class of approximate infinite dimensional optimal control problems. Proc. American Control Conference (Denver CO), 2020.
• 5. V. Basco and P.M. Dower. A two-player representation for a class of infinite horizon control problems under state constraints. Proc. American Control Conference (Denver CO), 2020.
• 6. W.M. McEneaney and P.M. Dower. Conversion of a class of stochastic control problems to fundamental-solution deterministic control problems. Proc. American Control Conference (Denver CO), 2020.
• 7. A.I. Maass, D. Nesic, R. Postoyan, and P.M. Dower. Observer design for nonlinear networked control systems with persistently exciting protocols. IEEE Transactions on Automatic Control, 2019.
• 8. A.I. Maass, D. Nesic, R. Postoyan, and P.M. Dower. Lp stability of networked control systems implemented on WirelessHART. Automatica, 2019.
• 9. W.M. McEneaney and P.M. Dower. Staticization-based representations for Schrodinger equa- tions driven by coulomb potentials. Proc. 3rd IFAC workshop on Thermodynamic Foundations for a Mathematical Systems Theory (Louvain-la-Neuve, Belgium), 2019.
• 10. J. Kennedy, A. Chapman, and P.M. Dower. Generalised coverage control for time-varying density functions. Proc. European Control Conference (Napoli, Italy), 2019.
• 11. P.M. Dower, W.M. McEneaney, and I. Yegorov. Exploiting characteristics in stationary action problems. Proc. SIAM Conference on Control & Its Applications (Chengdu, China), 2019.
• 12. P.M. Dower. An adaptive max-plus eigenvector method for continuous time optimal control problems. Numerical methods for optimal control problems, INDAM Series, vol. 29, Springer (Eds. M. Falcone, R. Ferretti, L. Grune, W. McEneaney), 2018.
• 13. F. Taringoo, P.M. Dower, D. Nesic, and Y. Tan. Optimization methods on Riemannian manifolds via extremum seeking algorithms. SIAM J. Control & Optimization, 56(5):3867–3892, 2018.
• 14. I. Yegorov and P.M. Dower. Perspectives on characteristics based curse-of-dimensionality-free numerical approaches to solving Hamilton Jacobi equations. Applied Math. & Optim., DOI:10.1007/s00245-018-9509-6, 2018.
• 15. I. Yegorov, P. Dower, L. Grune. Global extension of local control Lyapunov functions via exit-time optimal control. Proc. IEEE Conference on Decision & Control (Miami, Fl), 2018.
• 16. A.I. Maass, D. Nesic, R. Postoyan, and P.M. Dower. Observer design for networked control systems implemented over WirelessHART. Proc. IEEE Conference on Decision & Control (Miami, Fl), 2018.
• 17. P.M. Dower. Basis adaptation for a max-plus eigenvector method arising in optimal control. Proc. 23rd International Symposium on Mathematical Theory of Networks and Systems (Hong Kong), pp.350-355, 2018.
• 18. I. Yegorov, P.M. Dower, and L. Grune. A characteristics based curse-of-dimensionality-free approach for approximating control Lyapunov functions and feedback stabilization. Proc. 23rd International Symposium on Mathematical Theory of Networks and Systems (Hong Kong), pp.342–349, 2018.
• 19. P.M. Dower and W.M. McEneaney. An action principle for constructing fundamental solution groups for wave equations. Proc. 23rd International Symposium on Mathematical Theory of Networks and Systems (Hong Kong), pp.702–704, 2018.
• 20. W.M. McEneaney and P.M. Dower. Static duality and a stationary action application. J. Differential Equations, 264(2):525-549, 2018.
• 21. G. Shi, B. Li, Z. Miao, P.M. Dower, and M.R. James. Reaching agreement in quantum hybrid networks”. Scientific Reports, 7(5989):1-9, 2017. 
• 22. P.M. Dower and M. Cantoni. State constrained optimal control of linear time-varying systems. Proc. IEEE Conference on Decision & Control (Melbourne, Australia), pp.1338-1343, 2017.
• 23. A.I. Maass, D. Nesic, R. Postoyan, P.M. Dower, and V.S. Varma. Emulation-based stabilization of networked control systems over WirelessHART. Proc. IEEE Conference on Decision & Control (Melbourne, Australia), pp.6628-6633, 2017.
• 24. P.M. Dower and W.M. McEneaney. Solving two-point boundary value problems for a wave equation via the principle of stationary action and optimal control. SIAM J. Control & Optimization, 55(4):2151-2205, 2017.
• 25. P.M. Dower and H. Zhang. A new fundamental solution for a class of differential Riccati equations. J. Math. Contr. Sig. Sys., 29(3):1-33, 2017.
• 26. C.M. Kellett, P.M. Dower, and H. Ito. Subclasses of integral input-to-state stability for in- terconnections. IEEE Transactions on Automatic Control, 62(5):2476-2482, 2017.
• 27. D.N. Tran, C.M. Kellett, and P.M. Dower. Qualitative equivalences of ISS and l2-gain stability properties for discrete-time nonlinear systems. Automatica, 77:360-369, 2017.
• 28. W.M. McEneaney and P.M. Dower. Staticization, its dynamic program, and solution propagation. Automatica, 81:56-67, 2017.
• 29. P.M. Dower and W.M. McEneaney. Representation of fundamental solution groups for wave equations via stationary action and optimal control.  Proc. IEEE American Control Conference (Seattle WA, USA), 2017.
• 30. P.M. Dower, W.M. McEneaney, and M. Cantoni. A game representation for state constrained linear regulator problems. Proc. 55th IEEE Conference on Decision & Control (Las Vegas NV, USA), pages 1074-1079, 2016. 
• 31. D.N. Tran, C.M. Kellett, and P.M. Dower. Analysis of discrete-time nonlinear l2-gain bounds via dynamic programming. Proc. Australian Control Conference (Newcastle, Australia), pages 66-70, 2016. 
• 32. G. Shi, B. Li, Z. Miao, P.M. Dower, and M.R. James. Reaching agreement in quantum hybrid networks. Proc. Australian Control Conference (Newcastle, Australia), pages 159-161, 2016. 
• 33. Z. Miao and P.M. Dower. A reduced complexity min-plus method for synchronization of qubits. Proc. Australian Control Conference (Newcastle, Australia), pages 391-396, 2016. 
• 34. A.I. Maass, D. Nesic, and P.M. Dower. A hybrid model of networked control systems imple- mented on wirelesshart networks under source routing configuration. Proc. Australian Control Conference (Newcastle, Australia), pages 60-65, 2016. 
• 35. P.M. Dower and M. Cantoni. An approximating game for a continuous-time state-constrained linear regulator problem. Proc. Australian Control Conference (Newcastle, Australia), pages 197-202, 2016.
• 36. P.M. Dower and W.M. McEneaney. On existence and uniqueness of stationary action trajectories.  Proc. MTNS, pages 624-631, 2016.
• 37. P.M. Dower, W.M. McEneaney, and M. Cantoni. A dynamic game approximation for a linear regulator problem with a log-barrier state constraint. Proc. MTNS, pages 297-204, 2016.
• 38. W.M. McEneaney and P.M. Dower. The Hamilton-Jacobi equation corresponding to complex-valued stationary-action problems. Proc. MTNS, pages 319-326, 2016.
• 39. C.M. Kellett and P.M. Dower. Input-to-state stability, integral input-to-state stability, and L2-gain properties: qualitative equivalences and interconnect systems. IEEE Transactions on Automatic Control, 61(1):3-17, 2016.
• 40. W.M. McEneaney and P.M. Dower. The principle of least action and fundamental solutions of mass-spring and N-body two-point boundary value problems. SIAM J. Control & Optimization (preprint), 53(5):2898-2933, 2015.
• 41. P.M. Dower and W.M. McEneaney. A max-plus dual space fundamental solution for a class of operator differential Riccati equations. SIAM J. Control & Optimization (preprint arXiv:1404.7209), 53(2): 969-1002, 2015.
• 42. H. Zhang and P.M. Dower. A max-plus based fundamental solution for a class of discrete time linear regulator problems. Linear Algebra and its Applications, 471:693-729, 2015.
• 43. H. Zhang and P.M. Dower. Max-plus fundamental solution semigroups for a class of difference Riccati equations. Automatica, 52:103-110, 2015.
• 44. P.M. Dower and W.M. McEneaney. An optimal control approach to the approximation of fundamental solution groups for lossless wave equations. Proc. IEEE Conference on Decision and Control (Osaka, Japan), pages 3882-3887, 2015.
• 45. F. Taringoo, D. Nesic, P.M. Dower, and Y. Tan. Coordination of blind agents on Lie groups. Proc. IEEE Conference on Decision and Control (Osaka, Japan), pages 7256-7261, 2015.
• 46. D.N. Tran, C.M. Kellett, and P.M. Dower. Input-to-state stability with respect to two measures: discrete-time systems. Proc. IEEE Conference on Decision and Control (Osaka, Japan), pages  1817-1822, 2015. 
• 47. D.N. Tran, C.M. Kellett, and P.M. Dower. A discrete-time strong implication-form Lyapunov function for ISS systems with respect to positive definite measurement functions. Proc. Australian Control Conference (Gold Coast, Australia), pages 39-42, 2015. 
• 48. P.M. Dower and H. Zhang. A new fundamental solution for differential Riccati equations arising in L2-gain analysis. Proc. Australian Control Conference (Gold Coast, Australia), pages 65-68, 2015. 
• 49. P.M. Dower and C.M. Kellett. A small-gain feedback interconnection for bilinear systems. Proc. Australian Control Conference (Gold Coast), pages 126-130, 2015.
• 50. D.N. Tran, C.M. Kellett, and P.M. Dower. Equivalences of stability properties for discrete-time nonlinear systems. Proc. IFAC MICNON’15, 782-787, 2015.
• 51. W.M. McEneaney and P.M. Dower. Staticization and associated Hamilton-Jacobi and Riccati equations. Proc. SIAM CT’15, pages 376-383, 2015.
• 52. P.M. Dower, W.M. McEneaney, and H. Zhang. Max-plus fundamental solution semigroups for optimal control problems. Proc. SIAM CT’15, pages 368-375, 2015.
• 53. P.M. Dower and W.M. McEneaney. A max-plus fundamental solution semigroup for a class of lossless wave equations. Proc. SIAM CT’15, pages 400-407, 2015.
• 54. P. M. Dower and C. M. Kellett. Nonlinear systems with nonlinear L2-gain. Proc. IEEE Conference on Decision and Control (Los Angeles CA, USA), 777-782, 2014. 
• 55. F. Taringoo, D. Nesic, Y. Tan, and P. M. Dower. Extremum seeking control for nonlinear systems on compact Riemannian manifolds. Proc. IEEE Conference on Decision and Control (Los Angeles CA, USA), pages 2667-2772, 2014. 
• 56. P.M. Dower and W.M. McEneaney. Max-plus fundamental solution semigroups for dual operator differential Riccati equations. Proc. Australian Control Conference (Canberra, Australia), pages 19-24, 2014. 
• 57. P.M. Dower and C.M. Kellett. Nonlinear L2-gain verification for bilinear systems. Proc. Australian Control Conference (Canberra, Australia), pages 227-232, 2014. 
• 58. H. Zhang and P.M. Dower. Pruning error analysis for a class of curse-of-dimensionality free methods. Proc. Australian Control Conference (Canberra, Australia), pages 176-181,2014. 
• 59. H. Zhang and P.M. Dower. A new non-iterative solution for a class of difference Riccati equations. Proc. Australian Control Conference (Canberra, Australia), pages 112-117, 2014.
• 60. P.M. Dower and W.M. McEneaney. A max-plus dual space fundamental solution semigroup for operator differential Riccati equations. Proc. 21st International Symposium on Mathematical Theory of Networks and Systems (Groningen, The Netherlands), pages 148-153, 2014. 
• 61. W.M. McEneaney and P.M. Dower. The principle of stationary action and numerical methods for n-body problems. Proc., 21st International Symposium on Mathematical Theory of Networks and Systems (Groningen, The Netherlands), pages 1442-1445, 2014.
• 62. H. Zhang and P.M. Dower. A max-plus primal space fundamental solution for a class of difference Riccati equations. Proc. 21st International Symposium on Mathematical Theory of Networks and Systems (Groningen, The Netherlands), pages 1446-1452, 2014. 
• 63. H. Zhang and P.M. Dower. Analysis of difference Riccati equations via a new max-plus based fundamental solution. Proc. 21st International Symposium on Mathematical Theory of Net- works and Systems (Groningen, The Netherlands), pages 679–685, 2014.
• 64. P.M. Dower and W.M. McEneaney. A fundamental solution for an infinite dimensional two-point boundary value problem via the principle of stationary action. Proc. Australian Control Conference (Perth, Australia), 2013. 
• 65. H. Zhang, P.M. Dower, and W.M. McEneaney. A pruning algorithm for managing complexity in the solution of a class of linear non-quadratic regulator problems. Proc. Australian Control Conference (Perth, Australia), 2013.
• 66. C.M. Kellett and P.M. Dower. Stability of (integral) input-to-state stable interconnected nonlinear systems via qualitative equivalences. Proc. Australian Control Conference (Perth, Australia), 2013.
• 67. A. Farhadi, P.M. Dower, and M. Cantoni. Computation time analysis of centralized and distributed optimization algorithms applied to automated irrigation networks. Proc. Australian Control Conference (Perth, Australia), 2013.
• 68. F. Taringoo, D. Nesic, Y. Tan, and P.M. Dower. Closeness of solutions and averaging for nonlinear systems on Riemannian manifolds. Proc. Australian Control Conference (Perth, Australia), 2013.
• 69. A. Farhadi, M. Cantoni, and P.M. Dower. Computational complexity analysis of a consensus based distributed optimization method. Proc. IEEE Conference on Decision and Control (Florence, Italy), 2013.
• 70. F. Taringoo, D. Nesic, Y. Tan, and P.M. Dower. Averaging for nonlinear systems on Riemannian manifolds. Proc. IEEE Conference on Decision and Control (Florence, Italy), 2013.
• 71. C.M. Kellett, F. Wirth, and P.M. Dower. Input-to-state stability, integral input-to-state stability, and non-compact level sets. Proc. NOLCOS, 2013. 
• 72. W.M. McEneaney and P.M. Dower. The principle of least action and solution of two-point boundary value problems on a limited time horizon. Proc. SIAM Conference on Control and Its Applications (San Diego CA, USA), 2013.
• 73. H. Zhang and P.M. Dower. A max-plus method for the approximate solution of discrete-time linear regulator problems with non-quadratic terminal payoff. Proc. SIAM Conference on Control and Its Applications (San Diego CA, USA), 2013.
• 74. H. Zhang and P.M. Dower. Max-plus based computation of nonlinear L2-gain performance bounds using a piecewise affine-quadratic basis. Proc. 51st European Control Conference (Zurich, Switzerland), 2013.
• 75. H. Zhang and P.M. Dower. A max-plus based approach to the solution of a class of LQR prob- lems with non-quadratic terminal payoff. Proc. 51st European Control Conference (Zurich, Switzerland), 2013 
• 76. H. Zhang and P.M. Dower. Computation of tight integral input-to-state stability bounds for nonlinear systems. Systems and Control Letters, 62:355-365, 2013.
• 77. H. Zhang and P.M. Dower. Performance bounds for nonlinear systems with a nonlinear L2-gain property. Int. J. Control, 85(9):1293-1312, 2012.
• 78. P.M. Dower and W.M. McEneaney. A max-plus method for the optimal control of a diffusion equation. Proc. IEEE Conference on Decision and Control (Maui HI, USA), 2012.
• 79. H. Zhang and P.M. Dower. Approximation of nonlinear L2-gain bounds via a max-plus method. Proc. IEEE Conference on Decision and Control (Maui HI, USA), 2012.
• 80. P.M. Dower, C.M. Kellett, and H. Zhang. A weak L2-gain property for nonlinear systems. IEEE Conference on Decision and Control (Maui HI, USA), 2012.
• 81. C.M. Kellett and P.M. Dower. A generalization of input-to-state stability. Proc. IEEE Conference on Decision and Control (Maui HI, USA), 2012.
• 82. P.M. Dower. An approximation arising in max-plus based optimal stopping. Proc. Australian Control Conference (Sydney, Australia), 2012.
• 83. P.M. Dower and H. Zhang. Tight comparison function bounds for nonlinear systems with a nonlinear L2-gain property. Proc. Australian Control Conference (Sydney, Australia), 2012.
• 84. A. Farhadi, M. Cantoni and P.M. Dower. Performance and information pattern trade-offs in a consensus based distributed optimization method. Proc. Australian Control Conference (Sydney, Australia), 2012.
• 85. H. Zhang and P.M. Dower. An explicit solution to a class of constrained optimal control problems. Proc. Australian Control Conference (Sydney, Australia), 2012.
• 86. H. Zhang and P.M. Dower. An improved max-plus eigenvector method for the approximation of nonlinear L2-gain bounds. Proc. MTNS (Melbourne, Australia), 2012.
• 87. P.M. Dower, H. Zhang, and C.M. Kellett, Nonlinear L2-gain verification for nonlinear systems. Systems and Control Letters, 61(4):563-572, 2012.
• 88. P.M. Dower and W.M. McEneaney. A max-plus based fundamental solution for a class of infinite dimensional Riccati equations. Proc. IEEE Conference on Decision and Control and European Control Conference (Orlando FL, USA), 2011.
• 89. P.M. Dower and H. Zhang. Deterministic optimal stopping via a max-plus method. Proc. Australian Control Conference (Melbourne), 2011. 
• 90. P.M. Dower, H. Zhang, and C.M. Kellett. A verification theorem for nonlinear systems with nonlinear L-gain. Proc. Australian Control Conference (Melbourne), 2011. 
• 91. H. Zhang and P.M. Dower. Minimum integral input-to-state stability bounds. Proc. Australian Control Conference (Melbourne), 2011. 
• 92. M. Kearney, P.M. Dower, and M. Cantoni. Model predictive control for flood mitigation: A Wivenhoe dam case study. Proc. Australian Control Conference (Melbourne), 2011.
• 93. M. Kearney, M. Cantoni, and P.M. Dower. Non-iterative distributed MPC for large-scale irrigation channels. Proc. Australian Control Conference (Melbourne, Australia), 2011.
• 94. H. Zhang, P.M. Dower, and C.M. Kellett. State feedback controller synthesis to achieve a nonlinear L2-gain property. Proc. 18th IFAC World Congress (Milano, Italy), 2011.
• 95. M. Kearney, M. Cantoni, and P.M. Dower. Model predictive control for systems with scheduled load and its application to automated irrigation channels. Proc. IEEE Conference on Networking, Sensing & Control (Delft, The Netherlands), 2011.
• 96. P.M. Dower, H. Zhang, and C.M. Kellett. Nonlinear L2-gain analysis via a cascade. Proc. IEEE Conference on Decision and Control (Atlanta GA, USA), 2010.
• 97. H. Zhang, P.M. Dower, and C.M. Kellett. A bounded real lemma for nonlinear L2-gain. Proc. IEEE Conference on Decision and Control (Atlanta GA, USA), 2010.
• 98. B.S. Rueffer, H. Ito, and P.M. Dower. Computing asymptotic gains of large-scale interconnections. Proc. IEEE Conference on Decision and Control (Atlanta GA, USA), 2010.
• 99. P.M. Dower and P.M. Farrell. Small-signal power transient modelling and regulation in Raman amplified optical fibre links. Int. J. Control, 2010.
• 100. B.S. Rueffer, C.M. Kellett, P.M. Dower, and S.R. Weller. Belief propagation as a dynamical system: The linear case and open problems. IET Control Theory and Applications, 2010.
• 101. H. Zhang and P.M. Dower. A max-plus method for the approximation of transient bounds for systems with nonlinear L2-gain. Proc. MTNS (Budapest, Hungary), 2010.
• 102. B.S. Rueffer, C.M. Kellett, and P.M. Dower. On copositive Lyapunov functions for a class of monotone systems. Proc. MTNS (Budapest, Hungary), 2010.
• 103. B.S. Rueffer, P.M. Dower, C.M. Kellett, S.R. Weller. On robust stability of belief propagation algorithms for LDPC decoding. Proc. MTNS (Budapest, Hungary), 2010.
• 104. B.S. Rueffer, P.M. Dower, and H. Ito. Computational comparison principles in large-scale system stability analysis. Proc. 10th Annual Conference on Control Systems (Kumamoto, Japan), 2010.