Stability notions for a class of nonlinear systems with measure controls

We consider the problem of stability in a class of differential equations which are driven by a differential measure associated with inputs of locally bounded variation. After discussing some existing notions of solution for such systems, we derive Lyapunov-based conditions on the system’s vector fields for asymptotic stability under a specific class of inputs. These conditions are based on the stability margin of the Lebesgue-integrable and the measure-driven components of the system. For more general inputs which do not necessarily lead to asymptotic stability, we then derive conditions such that the maximum norm of the resulting trajectory is bounded by some function of the total variation of the input, which generalizes the notion of integral input-to-state stability in measure-driven systems.

[1]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[2]  P. Olver.  Nonlinear Systems , 2013 .

[3]  Bernard Brogliato,et al.  Stability and Observer Design for Lur'e Systems with Multivalued, Nonmonotone, Time-Varying Nonlinearities and State Jumps , 2014, SIAM J. Control. Optim..

[4]  Eduardo Sontag.  Comments on integral variants of ISS , 1998 .

[5]  A. Bressan.  On differential systems with impulsive controls , 1987 .

[6]  Geraldo Nunes Silva,et al.  Closed loop stability of measure-driven impulsive control systems , 2010 .

[7]  João Pedro Hespanha,et al.  Lyapunov conditions for input-to-state stability of impulsive systems , 2008, Autom..

[8]  Alberto Bressan,et al.  On differential systems with quadratic impulses and their applications to Lagrangian mechanics , 1993 .

[9]  V. Lakshmikantham,et al.  Stability criteria for impulsive differential equations in terms of two measures , 1989 .

[10]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[11]  Eduardo Sontag.  Smooth stabilization implies coprime factorization , 1989, IEEE Transactions on Automatic Control.

[12]  Thomas Carter,et al.  Optimal impulsive space trajectories based on linear equations , 1991 .

[13]  Daizhan Cheng,et al.  Optimal impulsive control in periodic ecosystem , 2006, Syst. Control. Lett..

[14]  O. Hájek.  Review: S. G. Pandit and S. G. Deo, Differential systems involving impulses , 1985 .

[15]  Vincent Acary,et al.  Higher order Moreau’s sweeping process: mathematical formulation and numerical simulation , 2008, Math. Program..

[16]  Xinzhi Liu,et al.  Input-to-state stability of impulsive and switching hybrid systems with time-delay , 2011, Autom..

[17]  David Angeli,et al.  A characterization of integral input-to-state stability , 2000, IEEE Trans. Autom. Control..

[18]  R. Hartl,et al.  Dynamic Optimal Control Models in Advertising: Recent Developments , 1994 .

[19]  Y. Orlov.  Schwartz' distributions in nonlinear setting: Applications to differential equations, filtering and optimal control , 2002 .

[20]  A. Bressan,et al.  Impulsive control systems without commutativity assumptions , 1994 .

[21]  L. Neustadt.  A general theory of minimum-fuel space trajectories : technical report , 1964 .

[22]  S. G. Pandit,et al.  Differential systems involving impulses , 1982 .

[23]  S. Leela,et al.  Stability of measure differential equations. , 1974 .

[24]  Denis Arzelier,et al.  Measures and LMI for impulsive optimal control with applications to space rendezvous problems , 2012, 2012 American Control Conference (ACC).

[25]  Ke Wang,et al.  Optimal impulsive harvesting policy for single population , 2003 .

[26]  H. Sussmann.  On the Gap Between Deterministic and Stochastic Ordinary Differential Equations , 1978 .

[27]  Stephan Trenn,et al.  Regularity of distributional differential algebraic equations , 2009, Math. Control. Signals Syst..

[28]  J. Warga,et al.  Variational Problems with Unbounded Controls , 1965 .

[29]  Panagiotis D. Christofides,et al.  Stability of nonlinear asynchronous systems , 2007, 2007 46th IEEE Conference on Decision and Control.

[30]  Aneel Tanwani,et al.  An observer for switched differential-algebraic equations based on geometric characterization of observability , 2013, 52nd IEEE Conference on Decision and Control.

[31]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[32]  SERGEY DASHKOVSKIY,et al.  Input-to-State Stability of Nonlinear Impulsive Systems , 2012, SIAM J. Control. Optim..

[33]  P. C. Das,et al.  Existence and stability of measure differential equations , 1972 .

[34]  van de N Nathan Wouw,et al.  Robust impulsive control of motion systems with uncertain friction , 2012 .

[35]  G. D. Maso,et al.  On systems of ordinary differential equations with measures as controls , 1991, Differential and Integral Equations.

[36]  M. Marques,et al.  Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction , 1993 .

[37]  W. Schmaedeke.  Optimal Control Theory for Nonlinear Vector Differential Equations Containing Measures , 1965 .

[38]  H. Schaub,et al.  Impulsive Feedback Control to Establish Specific Mean Orbit Elements of Spacecraft Formations , 2001 .

[39]  A. Bressan,et al.  Impulsive control systems with commutative vector fields , 1991 .

[40]  J. Moreau,et al.  A chain rule involving vector functions of bounded variation , 1987 .

[41]  Raymond W. Rishel,et al.  An Extended Pontryagin Principle for Control Systems whose Control Laws Contain Measures , 1965 .

[42]  B. Brogliato.  Some results on optimal control with unilateral state constraints , 2009 .

[43]  B. Miller.  The generalized solutions of nonlinear optimization problems with impulse control , 1996 .

[44]  Š. Schwabik,et al.  Generalized Ordinary Differential Equations , 1992 .

[45]  R. Vinter,et al.  Necessary conditions for optimal impulsive control problems , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.