This paper is concerned with the optimal boundary control of a non-dimensional non-linear parabolic system consisting of the Kuramoto–Sivashinsky–Korteweg–de Vries equation and a heat equation. By the Dubovitskii and Milyutin functional analytical approach, first in the fixed final horizon case we prove the Pontryagin maximum principle of the optimal control problem of this coupled system. Then under weaker additional conditions, we study the controlled system in the free final horizon case and present further investigational results of current interests. The necessary optimality conditions are established for optimal control problems in these two cases. Finally, a remark on how to utilize the obtained results is also made for illustration.
AnnunziatoMPizzutiSTsimringLS (2000) Analysis and prediction of spatio-temporal flame dynamics. In: Proceedings of the IEEE workshop on nonlinear dynamics of electronic systems (eds SettiGRovattiRMazziniG), Catania, Italy, 18–20 May 2000, pp.117–121. Singapore: World Scientific.
2.
ArmaouAChristofidesPD (2002) Dynamic optimization of dissipative PDE systems using nonlinear order reduction. Chemical Engineering Science57: 5083–5114.
3.
BrysonAEJr (1996) Optimal control 1950 to 1985. IEEE Control Systems Magazine16: 26–33.
4.
CerpaE (2014) Boundary control of Korteweg-de Vries and Kuramoto–Sivashinsky PDEs. In: SamadTBaillieulJ (eds) Encyclopedia of Systems and Control. London: Springer-Verlag, pp.1–8.
5.
CerpaEMercadoAPazotoAF (2012) On the boundary control of a parabolic system coupling KS–KdV and heat equations. SCIENTIA, Series A: Mathematical Sciences22: 55–74.
6.
ChanWLGuoBZ (1990) Optimal birth control of population dynamics. II. Problems with free final time, phase constraints, and mini-max costs. Journal of Mathematical Analysis and Applications146: 523–539.
7.
ChanWLGuoBZ (1989) Optimal birth control of population dynamics. Journal of Mathematical Analysis and Applications144: 532–552.
8.
ChristofidesPDArmaouA (2000) Global stabilization of the Kuramoto–Sivashinsky equation via distributed output feedback control. Systems and Control Letters39: 283–294.
9.
FattoriniHO (2005) Infinite Dimensional Linear Control Systems. The Time Optimal and Norm Optimal Problems. North-Holland Mathematics Studies 201. Amsterdam: Elsevier Science B.V.
10.
GayteIGuillén-GonzálezFRojas-MedarM (2010) Dubovitskii-Milyutin formalism applied to optimal control problems with constraints given by the heat equation with final data. IMA Journal of Mathematical Control and Information27: 57–76.
11.
GibsonJALowingerJF (1974) A predictive min-H method to improve convergence to optimal solutions. International Journal of Control19: 575–592.
12.
GirsanovIV (1972) Lectures on Mathematical Theory of Extremum Problems. Lecture Notes in Economics and Mathematical Systems 67. Berlin: Springer-Verlag.
13.
HeJWGlowinskiRGormanMet al. (1998) Some results on the controllability and the stabilization of the Kuramoto–Sivashinsky equation. In: LionsJL (ed) Équations aux Dérivées Partielles et Applications. Paris: Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, pp.571–590.
14.
HoYCPepyneDL (2002) Simple explanation of the no-free-lunch theorem and its implications. Journal of Optimization Theory and Applications115: 549–570.
15.
HuntleyE (1985) Optimal boundary control of a tracking problem for a parabolic distributed parameter system. International Journal of Control42: 411–431.
16.
KotarskiW (1997) Some Problem of Optimal and Pareto Optimal Control for Distributed Parameter Systems. Katowice: Wydawinictwo Uniwersytetu Śląskiego.
17.
KrstićMSmyshlyaevA (2008) Boundary Control of PDEs. A Course on Backstepping Designs. Advances in Design and Control 16. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM).
18.
KulkarniKZhangLLinningerAA (2006) Model and parameter uncertainty in distributed systems. Industrial and Engineering Chemistry Research45: 7832–7840.
19.
KuramotoYTsuzukiT (1975) On the formation of dissipative structures in reaction diffusion systems: Reductive perturbation approach. Progress of Theoretical Physics54: 687–699.
20.
LeeCHTranHT (2005) Reduced-order-based feedback control of the Kuramoto–Sivashinsky equation. Journal of Computational and Applied Mathematics173: 1–19.
21.
LenburyYRattanakulCLiD (2006) Stability of solution of Kuramoto–Sivashinsky–Korteweg–de Vries system. Computers and Mathematics with Applications52: 497–508.
22.
LiXYongJ (1995) Optimal Control Theory for Infinite Dimensional Systems. Boston: Birhäuser.
23.
LiuWJKrstićM (2001) Stability enhancement by boundary control in the Kuramoto–Sivashinsky equation. Nonlinear Analysis: Theory, Methods and Applications43: 485–507.
MatarOKCrasterRVKumarS (2007) Falling films on flexible inclines. Physical Review E76: 056301.
26.
OrlovYV (1988) Theory of Optimal Systems with Generalized Controls. Moscow: ‘Nauka’ (in Russian).
27.
PassenbergBLeiboldMStursbergOet al. (2014) A globally convergent, locally optimal min-H algorithm for hybrid optimal control. SIAM Journal on Control and Optimization52: 718–746.
28.
PourkargarDBArmaouA (2015) Design of APOD-based switching dynamic observers and output feedback control for a class of nonlinear distributed parameter systems. Chemical Engineering Science136: 62–75.
29.
PourkargarDBArmaouA (2013) Modification to adaptive model reduction for regulation of distributed parameter systems with fast transients. AIChE Journal59: 4595–4611.
30.
RamaswamyBChippadaSJooSW (1996) A full-scale numerical study of interfacial instabilities in thin-film flows. Journal of Fluid Mechanics325: 163–194.
31.
RebiaiSEZinoberASI (1993) Stabilization of uncertain distributed parameter systems. International Journal of Control57: 1167–1175.
32.
SivashinskyGI (1977) Nonlinear analysis of hydrodynamic instability in laminar flames, I. Derivation of basic equations. Acta Astronautica4: 1177–1206.
33.
SlossJMBruchJCJrSadekISet al. 1998: Maximum principle for optimal boundary control of vibrating structures with applications to beams. Dynamics and Control8: 355–375.
34.
von StrykOBulirschR (1992) Direct and indirect methods for trajectory optimization. Annals of Operations Research37: 357–373.
35.
SunB (2010) Maximum principle for optimal boundary control of the Kuramoto–Sivashinsky equation. Journal of the Franklin Institute347: 467–482.
36.
TröltzschF (2010) Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Graduate Studies in Mathematics 112. Translated from the 2005 German original by Jürgen Sprekels, Providence, RI: American Mathematical Society.
37.
XingJZhangCXuH (2003) Basics of Optimal Control Application. Beijing: Science Press (in Chinese).
38.
YangYDubljevicS (2013) Boundary model predictive control of thin film thickness modelled by the Kuramoto–Sivashinsky equation with input and state constraints. Journal of Process Control23: 1362–1379.
