In this paper we propose a convex programming based method to address a long-standing problem of inner-approximating backward reachable sets of state-constrained polynomial systems subject to time-varying uncertainties. The backward reachable set is a set of states, from which all trajectories starting will surely enter a target region at the end of a given time horizon without violating a set of state constraints in spite of the actions of uncertainties. It is equal to the zero sub-level set of the unique Lipschitz viscosity solution to a Hamilton-Jacobi partial differential equation (HJE). We show that inner-approximations of the backward reachable set can be formed by zero sub-level sets of its viscosity super-solutions. Consequently, we reduce the inner-approximation problem to a problem of synthesizing polynomial viscosity super-solutions to this HJE. Such a polynomial solution in our method is synthesized by solving a single semi-definite program. We also prove that polynomial solutions to the formulated semi-definite program exist and can produce a convergent sequence of inner-approximations to the interior of the backward reachable set in measure under appropriate assumptions. This is the main contribution of this work. Several illustrative examples demonstrate the merits of our approach.
Related WorkAs mentioned above, inner-approximate reachability analysis of ordinary differential equations subject to time-varying uncertainties and state constraints, is still in its infancy and thus provides an open area of research.For ordinary differential equations free of time-varying uncertainties and state constraints, [17], [18] proposed a method based on modal intervals with affine forms to inner-approximate reachable sets using intervals. By making use of the homeomorphism property of the solution mapping, a boundary based reachability analysis method was proposed to inner-approximate reachable sets with polytopes in [44], and it was extended to a class of delay differential equations in [43]. Since reachable sets of nonlinear systems tend to be non-convex, the above mentioned methods based on convex set representations may result in poor approximations. As accuracy is also an important factor in performing reachability analysis (e.g.,[37], [28]), more complex shapes of representations such as Taylor models and semi-algebraic sets are desirable.[7] proposed a Taylor model backward flowpipe method to compute inner-approximations. [41] proposed an iterative method, with each iteration relying on solving semi-definite programming problems, to compute semi-algebraic inner-approximations for polynomial systems using the advection map of the given dynamical system. [45] extended the method in [44] to compute semi-algebraic inner-approximations of reachable sets for polynomial systems and beyond by solving semi-definite programming problems. Recently, [42] formulated the problem of solving HJEs as a semi-definite program to compute inner-approximations for polynomial systems. For state-constrained polynomial systems ...
