Some decidable results on reachability of solvable systems

Reachability analysis plays an important role in verifying the safety of modern control systems. In the existing work, there are many decidable results on reachability of discrete systems. For continuous systems, however, the known decidable results are established merely for linear systems. In this paper, we propose a class of nonlinear systems (named solvable systems) extending linear systems. We first show that their solutions are of closed form. On the basis of it, we study a series of reachability problems for various subclasses of solvable systems. Our main results are that these reachability problems are decidable by manipulations in number theory, real root isolation, and quantifier elimination. Finally the decision procedures are implemented in a Maple package REACH to solve several non-trivial examples.

[1]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[2]  Thomas A. Henzinger,et al.  The Algorithmic Analysis of Hybrid Systems , 1995, Theor. Comput. Sci..

[3]  Zhenbing Zeng,et al.  Reachability analysis of rational eigenvalue linear systems , 2010, Int. J. Syst. Sci..

[4]  Stefan Ratschan,et al.  Providing a Basin of Attraction to a Target Region of Polynomial Systems by Computation of Lyapunov-Like Functions , 2010, SIAM J. Control. Optim..

[5]  Mukarram Ahmad,et al.  Continued fractions , 2019, Quadratic Number Theory.

[6]  Robert K. Brayton,et al.  Model-checking continuous-time Markov chains , 2000, TOCL.

[7]  Zohar Manna,et al.  Verifying Hybrid Systems , 1992, Hybrid Systems.

[8]  Naijun Zhan,et al.  Computing semi-algebraic invariants for polynomial dynamical systems , 2011, 2011 Proceedings of the Ninth ACM International Conference on Embedded Software (EMSOFT).

[9]  Josef Stoer,et al.  Numerische Mathematik 1 , 1989 .

[10]  Pravin Varaiya,et al.  What's decidable about hybrid automata? , 1995, STOC '95.

[11]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[12]  Bruno Buchberger,et al.  Computer algebra symbolic and algebraic computation , 1982, SIGS.

[13]  Pin-Lin Liu.  Delay-dependent stabilization for linear time-delay uncertain systems with saturating actuators , 2011, Int. J. Gen. Syst..

[14]  Ashish Tiwari,et al.  Nonlinear Systems: Approximating Reach Sets , 2004, HSCC.

[15]  Godfrey H. Hardy,et al.  An introduction to the theory of numbers (5. ed.) , 1995 .

[16]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[17]  Edmund M. Clarke,et al.  Computing differential invariants of hybrid systems as fixedpoints , 2008, Formal Methods Syst. Des..

[18]  Thomas Sturm,et al.  REDLOG: computer algebra meets computer logic , 1997, SIGS.

[19]  Volker Weispfenning,et al.  Deciding polynomial-exponential problems , 2008, ISSAC '08.

[20]  Zidong Wang,et al.  Reliable stabilization of stochastic time-delay systems with nonlinear disturbances , 2005, Int. J. Gen. Syst..

[21]  Ian M. Mitchell,et al.  Overapproximating Reachable Sets by Hamilton-Jacobi Projections , 2003, J. Sci. Comput..

[22]  S. Liberty,et al.  Linear Systems , 2010, Scientific Parallel Computing.

[23]  Stephen Prajna.  Barrier certificates for nonlinear model validation , 2006, Autom..

[24]  Gerardo Lafferriere,et al.  Symbolic Reachability Computation for Families of Linear Vector Fields , 2001, J. Symb. Comput..

[25]  Guy W. Cherry.  Computer Algebra—Symbolic and Algebraic Computation. 2nd Edition (B. Buchberger, G. E. Collins and R. Loos, ed.) , 1986 .

[26]  Arnaldo Vieira Moura,et al.  Transcendental inductive invariants generation for non-linear differential and hybrid systems , 2012, HSCC '12.

[27]  P. Henrici,et al.  Circular arithmetic and the determination of polynomial zeros , 1971 .

[28]  A. Baker.  Transcendental Number Theory , 1975 .

[29]  Hirokazu Anai,et al.  Reach Set Computations Using Real Quantifier Elimination , 2001, HSCC.

[30]  John Lygeros,et al.  On mean square boundedness of stochastic linear systems with bounded controls , 2012, Syst. Control. Lett..

[31]  Enric Rodríguez-Carbonell,et al.  Generating all polynomial invariants in simple loops , 2007, J. Symb. Comput..

[32]  Amir Pnueli,et al.  Reachability Analysis of Dynamical Systems Having Piecewise-Constant Derivatives , 1995, Theor. Comput. Sci..

[33]  Enric Rodríguez-Carbonell,et al.  Automatic Generation of Polynomial Loop Invariants: Algebraic Foundations , 2004, ISSAC '04.