Complexity of elementary hybrid systems

In this paper, we consider simple classes of nonlinear systems and prove that basic questions related to their stability and controllability are either undecidable or computationally intractable (NP-hard). As a special case, we consider a class of "hybrid" systems in which the state space is partitioned into two halfspaces and the dynamics in each halfspace correspond to a different linear system.

[1]  Jeffrey D. Ullman,et al.  Formal languages and their relation to automata , 1969, Addison-Wesley series in computer science and information processing.

[2]  M. Paterson Unsolvability in 3 × 3 Matrices , 1970 .

[3]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[4]  Eduardo Sontag Nonlinear regulation: The piecewise linear approach , 1981 .

[5]  John N. Tsitsiklis,et al.  The Complexity of Markov Decision Processes , 1987, Math. Oper. Res..

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

[7]  I. Daubechies,et al.  Sets of Matrices All Infinite Products of Which Converge , 1992 .

[8]  Eduardo D. Sontag,et al.  Interconnected Automata and Linear Systems: A Theoretical Framework in Discrete-Time , 1996, Hybrid Systems.

[9]  Eduardo Sontag From linear to nonlinear: some complexity comparisons , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

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

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

[12]  Michael S. Branicky,et al.  Universal Computation and Other Capabilities of Hybrid and Continuous Dynamical Systems , 1995, Theor. Comput. Sci..

[13]  L. Gurvits Stability of discrete linear inclusion , 1995 .

[14]  Hava T. Siegelmann,et al.  On the Computational Power of Neural Nets , 1995, J. Comput. Syst. Sci..

[15]  O. Toker On the Algorithmic Unsolvability of Some Stability Problems for Discrete Event Systems , 1996 .

[16]  J. Tsitsiklis,et al.  Spectral quantities associated to pairs of matrices are hard, when not impossible, to compute and to approximate , 1996 .

[17]  Yuri V. Matiyasevich,et al.  Decision problems for semi-Thue systems with a few rules , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

[18]  Olivier Bournez,et al.  On the Computational Power of Dynamical Systems and Hybrid Systems , 1996, Theor. Comput. Sci..

[19]  John N. Tsitsiklis,et al.  When is a Pair of Matrices Mortal? , 1997, Inf. Process. Lett..