Symbolic reachability computation for families of linear vector fields (Q5945290)

The authors identify classes of linear control systems of the type \(\dot{x}(t)=Ax(t)+u(t)\), for which the finite time reachable set can be computed using symbolic computation tools by means of quantifier elimination. NEWLINENEWLINENEWLINEDepending on the eigenvalue structure of \(A\), admissible spaces of control functions are derived, described by linear combinations of basis functions. The main idea behind this construction is that -- after suitable changes of variables, if necessary -- the exponential terms vanish in the description of the reachable set, which eventually allows for the use of symbolic computation tools. NEWLINENEWLINENEWLINEProceeding this way, the following three classes of systems are derived: NEWLINENEWLINENEWLINE(i) \(A\) is nilpotent and \(u\) is a linear combination of polynomials \(t^k\) NEWLINENEWLINENEWLINE(ii) \(A\) is diagonalizable with real rational eigenvalues and \(u\) is a linear combination of exponentials \(e^{\mu t}\) satisfying a non-resonance condition NEWLINENEWLINENEWLINE(iii) \(A\) is diagonalizable with purely imaginary eigenvalues and \(u\) is a linear combination of \(\sin(\mu t)\) and \(\cos(\mu t)\) satisfying a non-resonance condition NEWLINENEWLINENEWLINEFor each of these classes, the results are illustrated by examples.