×

zbMATH — the first resource for mathematics

Well-posed systems – the LTI case and beyond. (English) Zbl 1296.93072
Summary: This survey is an introduction to well-posed Linear Time-Invariant (LTI) systems for non-specialists. We recall the more general concept of a system node, classical and generalized solutions of system equations, criteria for well-posedness, the subclass of regular linear systems, some of the available linear feedback theory. Motivated by physical examples, we recall the concepts of impedance passive and scattering passive systems, conservative systems and systems with a special structure that belong to these classes. We illustrate this theory by examples of systems governed by heat and wave equations. We develop local and global well-posedness results for LTI systems with nonlinear (in particular, bilinear) feedback, by extracting the abstract idea behind various proofs in the literature. We apply these abstract results to derive well-posedness results for the Burgers and Navier-Stokes equations.

MSC:
93C05 Linear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
93C20 Control/observation systems governed by partial differential equations
93B35 Sensitivity (robustness)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aalto, A.; Malinen, J., Compositions of passive boundary control systems, athematical Control and Related Fields, 3, 1-19, (2013) · Zbl 1272.47015
[2] Adams, R. A., Sobolev spaces, (1975), Academic Press New York · Zbl 0314.46030
[3] Ammari, K., Dirichlet boundary stabilization of the wave equation, Asymptotic Analysis, 30, 117-130, (2002) · Zbl 1020.35042
[4] Ammari, K.; Liu, Z.; Tucsnak, M., Decay rates for a beam with pointwise force and moment feedback, Mathematics of Control, Signals, and Systems, 15, 229-255, (2002) · Zbl 1042.93034
[5] Ammari, K.; Tucsnak, M., Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM. Control, Optimisation Calculus of Variations, 6, 361-386, (2001) · Zbl 0992.93039
[6] Apkarian, P.; Gahinet, P.; Becker, G., Self-scheduled \(\mathcal{H}_\infty\) control of linear parameter-varying systems: a design example, Automatica, 31, 1251-1261, (1995) · Zbl 0825.93169
[7] Arov, D. Z., Passive linear systems and scattering theory, (Picci, G.; Gilliam, D., Dynamical systems, control, coding, computer vision, (1999), Birkhäuser Basel), 27-44 · Zbl 0921.93005
[8] Arov, D. Z.; Nudelman, M. A., Passive linear stationary dynamical scattering systems with continuous time, Integral Equations and Operator Theory, 24, 1-45, (1996) · Zbl 0838.47004
[9] Ball, J. A.; Carroll, P. T.; Uetake, Y., Lax-Phillips scattering theory and well-posed linear systems: a coordinate-free approach, Mathematics of Control, Signals, and Systems, 20, 37-79, (2008) · Zbl 1145.93024
[10] Bensoussan, A.; Da Prato, G.; Delfour, M. C.; Mitter, S. K., Representation and control of infinite dimensional systems. vol. 1, (1992), Birkhäuser Boston
[11] Bounit, H.; Hadd, S., Regular linear systems governed by neutral fdes, Journal of Mathematical Analysis and Applications, 320, 836-858, (2006) · Zbl 1156.34062
[12] Bounit, H.; Idrissi, A., Time-varying regular bilinear systems, SIAM Journal on Control and Optimization, 47, 1097-1126, (2008) · Zbl 1167.93013
[13] Brooks, R. M.; Schmitt, K., The contraction mapping principle and some applications, (Electronic journal of differential equations. monograph, Vol. 9, (2009), Texas State University-San Marcos, Department of Mathematics San Marcos, TX) · Zbl 1203.47093
[14] Byrnes, C. I.; Gilliam, D. S.; Shubov, V. I.; Weiss, G., Regular linear systems governed by a boundary controlled heat equation, Journal of Dynamical and Control Systems, 8, 341-370, (2002) · Zbl 1010.93052
[15] Chai, S. G.; Guo, B. Z., Well-posedness and regularity of weakly coupled wave-plate equation with boundary control and observation, Journal of Dynamical and Control Systems, 15, 331-358, (2003) · Zbl 1203.93055
[16] Chai, S. G.; Guo, B. Z., Feedthrough operator for linear elasticity system with boundary control and observation, SIAM Journal on Control and Optimization, 48, 3708-3734, (2010) · Zbl 1202.35320
[17] Chai, S. G.; Guo, B. Z., Well-posedness and regularity of naghdi’s shell equation under boundary control, Journal of Differential Equations, 249, 3174-3214, (2010) · Zbl 1387.93062
[18] Curtain, R. F., Linear operator inequalities for strongly stable weakly regular linear systems, Mathematics of Control, Signals, and Systems, 14, 299-337, (2001) · Zbl 1114.93029
[19] Curtain, R. F.; Weiss, G., Well-posedness of triples of operators (in the sense of linear systems theory), (Kappel, F.; Kunisch, K.; Schappacher, W., Control and estimation of distributed parameter systems, (1989), Birkhäuser Basel), 41-59
[20] Curtain, R. F.; Weiss, G., Exponential stabilization of well-posed systems by colocated feedback, SIAM Journal on Control and Optimization, 45, 273-297, (2006) · Zbl 1139.93026
[21] Curtain, R. F.; Weiss, G.; Weiss, M., Coprime factorization for regular linear systems, Automatica, 32, 1519-1531, (1996) · Zbl 0870.93025
[22] Curtain, R. F.; Zwart, H. J., An introduction to infinite-dimensional linear systems theory, (1995), Springer Verlag New York · Zbl 0839.93001
[23] Engel, K.-J.; Nagel, R., One-parameter semigroups for linear evolution equations, (2000), Springer Verlag New York · Zbl 0952.47036
[24] Fefferman, C.L. (2000). Existence and smoothness of the Navier-Stokes equation, official Millenium problem statement of the Clay Mathematics Institute, available at http://www.claymath.org/sites/default/files/navierstokes.pdf.
[25] Fujita, H.; Kato, T., On the Navier-Stokes initial value problem. I, Archive for Rational Mechanics and Analysis, 16, 269-315, (1964) · Zbl 0126.42301
[26] Guo, B. Z.; Luo, Y. H., Controllability and stability of a second-order hyperbolic system with collocated sensor/actuator, Systems & Control Letters, 46, 45-65, (2002) · Zbl 0994.93021
[27] Guo, B. Z.; Shao, Z. C., Regularity of a Schrödinger equation with Dirichlet control and collocated observation, Systems & Control Letters, 54, 1135-1142, (2005) · Zbl 1129.35447
[28] Guo, B. Z.; Shao, Z. C., Regularity of an Euler-Bernoulli equation with Neumann control and collocated observation, Journal of Dynamical and Control Systems, 12, 405-418, (2006) · Zbl 1111.93033
[29] Guo, B. Z.; Zhang, X., The regularity of the wave equation with partial Dirichlet control and colocated observation, SIAM Journal on Control and Optimization, 44, 1598-1613, (2005) · Zbl 1134.35318
[30] Haak, B.; Kunstmann, P. C., Admissibility of unbounded operators and well-posedness of linear systems in Banach spaces, Integral Equations and Operator Theory, 55, 497-533, (2006) · Zbl 1138.93361
[31] Haak, B.; Kunstmann, P. C., On kato’s method for Navier-Stokes equations, Journal of Mathematical Fluid Mechanics, 11, 492-535, (2009) · Zbl 1262.35178
[32] Hadd, S., Unbounded perturbations of \(C_0\)-semigroups on Banach spaces and applications, Semigroup Forum, 70, 451-465, (2005) · Zbl 1074.47017
[33] Jacob, B.; Dragan, V.; Pritchard, A. J., Infinite dimensional time varying systems with nonlinear output feedback, Integral Equations Operator Theory, 22, 440-462, (1995) · Zbl 0839.93057
[34] Jacob, B.; Morris, K.; Trunk, C., Minimum-phase infinite-dimensional second-order systems, IEEE Transactions on Automatic Control, 52, 1654-1665, (2007) · Zbl 1366.93277
[35] Jacob, B.; Partington, J., Admissibility of control and observation operators for semigroups: a survey, Operator Theory: Advances and Applications, 149, 199-221, (2004) · Zbl 1083.93025
[36] Jacob, B.; Partington, J.; Pott, S., Conditions for admissibility of observation operators and boundedness of Hankel operators, Integral Equations Operator Theory, 47, 315-338, (2003) · Zbl 1046.47036
[37] Jacob, B.; Zwart, H. J., Counterexamples concerning observation operators for \(C_0\)-semigroups, SIAM Journal on Control and Optimization, 43, 137-153, (2004) · Zbl 1101.93042
[38] Jacob, B.; Zwart, H. J., Linear port-Hamiltonian systems on infinite-dimensional spaces, (2012), Birkhäuser Basel · Zbl 1254.93002
[39] Jayawardhana, B.; Logemann, H.; Ryan, E. P., Infinite-dimensional feedback systems: the circle criterion and input-to-state stability, Communications in Information and Systems, 8, 413-444, (2008) · Zbl 1168.93021
[40] Jayawardhana, B.; Weiss, G., State convergence of passive nonlinear systems with an \(L^2\) input, IEEE Transactions on Automatic Control, 54, 1723-1727, (2009) · Zbl 1367.93435
[41] Katsnelson, V.; Weiss, G., A counterexample in Hardy spaces with an application to systems theory, Zeitschrift für Analysis und ihre Anwendungen, 14, 705-730, (1995) · Zbl 0841.30031
[42] Krstic, M., On global stabilization of burgers’ equation by boundary control, Systems & Control Letters, 37, 123-141, (1999) · Zbl 1074.93552
[43] Lasiecka, I.; Triggiani, R., The operator \(B^\ast L\) for the wave equation with Dirichlet control, Abstract and Applied Analysis, 2004, 625-634, (2004) · Zbl 1065.35171
[44] Latushkin, Y.; Randolph, T.; Schnaubelt, R., Regularization and frequency-domain stability of well-posed systems, Mathematics of Control, Signals, and Systems, 17, 128-151, (2005) · Zbl 1110.93048
[45] Livšic, M. S., Operators, oscillations, waves (open systems), (1973), Amer. Math. Soc Providence, RI · Zbl 0254.47001
[46] Logemann, H.; Mawby, A. D., Low-gain integral control of infinite-dimensional regular linear systems subject to input hysteresis, (Colonius, F.; etal., Advances in mathematical systems theory, (2001), Springer-Verlag New York), 255-293, (Chapter 14)
[47] Logemann, H.; Rebarber, R.; Weiss, G., Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop, SIAM Journal on Control and Optimization, 34, 572-600, (1996) · Zbl 0853.93081
[48] Logemann, H.; Ryan, E. P., Systems with hysteresis in the feedback loop: existence, regularity and asymptotic behaviour of solutions, ESAIM Control, Optimisation and Calculus of Variations, 9, 169-196, (2003) · Zbl 1076.45004
[49] Ly, H. V.; Mease, K. D.; Titi, E. S., Distributed and boundary control of the viscous burgers’ equation, Numerical Functional Analysis and Optimization, 18, 143-188, (1997) · Zbl 0876.93045
[50] Malinen, J.; Staffans, O. J., Conservative boundary control systems, Journal of Differential Equations, 231, 290-312, (2006) · Zbl 1117.93025
[51] Malinen, J.; Staffans, O. J.; Weiss, G., When is a linear system conservative?, Quarterly of Applied Mathematics, 64, 61-91, (2006) · Zbl 1125.47007
[52] Morris, K. A., Justification of input-output methods for systems with unbounded control and observation, IEEE Transactions on Automatic Control, 44, 81-84, (1999) · Zbl 0989.93046
[53] Natarajan, V.; Gilliam, D.; Weiss, G., The state feedback regulator problem for regular linear systems, IEEE Transactions on Automatic Control, (2014), (in press) · Zbl 1360.93264
[54] Opmeer, M. R., Infinite-dimensional linear systems: a distributional approach, Proceedings of the London Mathematical Society, 91, 738-760, (2005) · Zbl 1087.47058
[55] Opmeer, M. R., Distribution semigroups and control systems, Journal of Evolution Equations, 6, 145-159, (2006) · Zbl 1114.93054
[56] Salamon, D., Infinite dimensional linear systems with unbounded control and observation: a functional analytic approach, Transactions of the American Mathematical Society, 300, 383-431, (1987) · Zbl 0623.93040
[57] Salamon, D., Realization theory in Hilbert space, Mathematical Systems Theory, 21, 147-164, (1989) · Zbl 0668.93018
[58] Schnaubelt, R., Feedbacks for nonautonomous regular linear systems, SIAM Journal on Control and Optimization, 41, 1141-1165, (2002) · Zbl 1028.93018
[59] Schnaubelt, R.; Weiss, G., Two classes of passive time-varying well-posed linear systems, Mathematics of Control, Signals, and Systems, 21, 265-301, (2010) · Zbl 1202.93063
[60] Sohr, H., The Navier-Stokes equations. an elementary functional analytic approach, (2001), Birkhäuser Basel · Zbl 0983.35004
[61] Sontag, E. D., Mathematical control theory: deterministic finite dimensional systems, (1990), Springer-Verlag New York · Zbl 0703.93001
[62] Staffans, O. J., \(J\)-energy preserving well-posed linear systems, Applied Mathematics and Computer Science, 11, 1361-1378, (2001) · Zbl 1008.93024
[63] Staffans, O. J., Passive and conservative continuous-time impedance and scattering systems. part I: well-posed systems, Mathematics of Control, Signals, and Systems, 15, 291-315, (2002) · Zbl 1158.93321
[64] Staffans, O. J., Passive and conservative infinite-dimensional impedance and scattering systems (from a personal point of view), (Mathematical systems theory in biology, communications, computation, and finance (Notre Dame, IN, 2002), IMA vol. math. appl., Vol. 134, (2003), Springer New York), 375-413 · Zbl 1156.93326
[65] Staffans, O. J., Well-posed linear systems, (2004), Cambridge University Press Cambridge, UK
[66] Staffans, O. J., On scattering passive system nodes and maximal scattering dissipative operators, Proceedings of the American Mathematical Society, 141, 1377-1383, (2013) · Zbl 1275.47077
[67] Staffans, O. J.; Weiss, G., Transfer functions of regular linear systems. part II: the system operator and the Lax-Phillips semigroup, Transactions of the American Mathematical Society, 354, 3229-3262, (2002) · Zbl 0996.93012
[68] Staffans, O. J.; Weiss, G., Transfer functions of regular linear systems. part III: inversions and duality, Integral Equations Operator Theory, 49, 517-558, (2004) · Zbl 1052.93032
[69] Staffans, O. J.; Weiss, G., A physically motivated class of scattering passive linear systems, SIAM Journal on Control and Optimization, 50, 3083-3112, (2012) · Zbl 1264.93090
[70] Takahashi, T.; Tucsnak, M., Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid, Journal of Mathematical Fluid Mechanics, 6, 53-77, (2004) · Zbl 1054.35061
[71] Titchmarsh, E. C., The theory of functions, (1939), Oxford University Press London · JFM 65.0302.01
[72] Tucsnak, M.; Weiss, G., How to get a conservative well-posed linear system out of thin air. part II. controllability and stability, SIAM Journal on Control and Optimization, 42, 907-935, (2003) · Zbl 1125.93383
[73] Tucsnak, M.; Weiss, G., Observation and control for operator semigroups, (2009), Birkhäuser Verlag Basel · Zbl 1188.93002
[74] van der Schaft, A. J., \(L_2\)-gain and passivity techniques in nonlinear control, (1999), Springer-Verlag New York
[75] Weiss, G., Admissibility of unbounded control operators, SIAM Journal on Control and Optimization, 27, 527-545, (1989) · Zbl 0685.93043
[76] Weiss, G., Admissible observation operators for linear semigroups, Israel Journal of Mathematics, 65, 17-43, (1989) · Zbl 0696.47040
[77] Weiss, G., The representation of regular linear systems on Hilbert spaces, (Kappel, F.; Kunisch, K.; Schappacher, W., Control and estimation of distributed parameter systems (Vorau, 1988), Internat. ser. numer. math., Vol. 91, (1989), Birkhäuser Basel), 401-416
[78] Weiss, G., Regular linear systems with feedback, Mathematics of Control, Signals, and Systems, 7, 23-57, (1994) · Zbl 0819.93034
[79] Weiss, G., Transfer functions of regular linear systems. part I: characterizations of regularity, Transactions of the American Mathematical Society, 342, 827-854, (1994) · Zbl 0798.93036
[80] Weiss, G., Optimal control of systems with a unitary semigroup and with colocated control and observation, Systems & Control Letters, 48, 329-340, (2003) · Zbl 1157.49307
[81] Weiss, G.; Curtain, R. F., Dynamic stabilization of regular linear systems, IEEE Transactions on Automatic Control, 42, 4-21, (1997) · Zbl 0876.93074
[82] Weiss, G.; Staffans, O. J., Maxwell’s equations as a scattering passive linear system, SIAM Journal on Control and Optimization, 51, 3722-3756, (2013) · Zbl 1417.93157
[83] Weiss, G.; Staffans, O. J.; Tucsnak, M., Well-posed linear systems—a survey with emphasis on conservative systems, Applied Mathematics and Computer Science, 11, 101-127, (2001)
[84] Weiss, G.; Tucsnak, M., How to get a conservative well-posed linear system out of thin air. part I. well-posedness and energy balance, ESAIM. Control, Optimisation and Calculus of Variations, 9, 247-273, (2003) · Zbl 1063.93026
[85] Weiss, G.; Zhao, X., Well-posedness and controllability of a class of coupled linear systems, SIAM Journal on Control and Optimization, 48, 2719-2750, (2009) · Zbl 1203.93029
[86] Wen, R.; Chai, S.; Guo, B. Z., Well-posedness and exact controllability of the fourth order Schrödinger equation with boundary control and collocated observation, SIAM Journal on Control and Optimization, 52, 365-396, (2014) · Zbl 1292.93040
[87] Willems, J. C., Dissipative dynamical systems, part I: general theory, part II: linear systems with quadratic supply rates, Archive for Rational Mechanics and Analysis, 45, 352-392, (1972)
[88] Xu, C. Z.; Weiss, G., Spectral properties of infinite-dimensional closed-loop systems, Mathematics of Control, Signals, and Systems, 17, 153-172, (2005) · Zbl 1102.93024
[89] Zwart, H. J.; Jacob, B.; Staffans, O. J., Weak admissibility does not imply admissibility for analytic semigroups, Systems & Control Letters, 48, 341-350, (2003) · Zbl 1157.93421
[90] Zwart, H. J.; Le Gorrec, Y.; Maschke, B.; Villegas, J., Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain, ESAIM: Control, Optim. and Calculus of Variations, 16, 1077-1093, (2010) · Zbl 1202.93064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.