Invariance Conditions for Nonlinear Dynamical Systems

Abstract

Recently, Horváth et al. (Appl Math Comput, submitted) proposed a novel unified approach to study, i.e., invariance conditions, sufficient and necessary conditions, under which some convex sets are invariant sets for linear dynamical systems. In this paper, by utilizing analogous methodology, we generalize the results for nonlinear dynamical systems. First, the Theorems of Alternatives, i.e., the nonlinear Farkas lemma and the S-lemma, together with Nagumo’s Theorem are utilized to derive invariance conditions for discrete and continuous systems. Only standard assumptions are needed to establish invariance of broadly used convex sets, including polyhedral and ellipsoidal sets. Second, we establish an optimization framework to computationally verify the derived invariance conditions. Finally, we derive analogous invariance conditions without any conditions.

Appendix

Theorem 15 ([5, 12]).

Let P ={ x | Gx ≤ b} be a polyhedron, where $G \in \mathbb{R}^{m\times n}$ and $b \in \mathbb{R}^{m}.$ Let the discrete system, given as in (1) , be linear, i.e., f(x) = Ax. Then P is an invariant set for the discrete system (1) if and only if there exists a matrix H ≥ 0, such that HG = GA and Hb ≤ b.

Proof.

We have that P is an invariant set for the linear system if and only if the optimal objective values of the following m linear optimization problems are all nonnegative:

$$\displaystyle{ \mathop{\mathrm{min}}\limits \{b_{i} - G_{i}^{T}Ax\,\vert \,Gx \leq b\}\ \ \ i \in \mathrm{ I}(m). }$$

(33)

Problems (33) are equivalent to

$$\displaystyle{ -b_{i} +\mathop{ \mathrm{max}}\limits \{G_{i}^{T}Ax\,\vert \,Gx \leq b\}\ \ \ i \in \mathrm{ I}(m). }$$

(34)

The duals of these linear optimization problems presented in (34) are for all i ∈ I(m)

$$\displaystyle{ \begin{array}{rl} - b_{i} +\mathop{ \mathrm{min}}\limits &\ b^{T}H_{i} \\ \text{s.t. }&G^{T}H_{i} = A^{T}G_{i} \\ &\ \ H_{i} \geq 0.\end{array} }$$

(35)

Due to the Strong Duality Theorem of linear optimization, see, e.g., [22], the primal and dual objective function values are equal at optimal solutions, thus G _i ^T Ax = b ^T H _i . As the optimal value of (33) is nonnegative for all i ∈ I(m), we have b _i − b ^T H _i  ≥ 0. Thus b ≥ Hb. The proof is complete.