Projected Dynamical Systems on Irregular, Non-Euclidean Domains for Nonlinear Optimization

Oblique Projected Dynamical Systems and Incremental Stability Under State Constraints

“…This proves the last statement. Hence, F is α-strongly and maximal P-monotone if and only if F : x → P 1 2 F(P − 1 2 x) is α-strongly and maximal monotone, which is a key property for the existence of solutions to (11) and (12) according to Theorem 1. Interestingly, we can transform (12) back to the original coordinates x = P − 1 2 x, which yieldsẋ…”

“…We will now describe oblique PDS, see also [11], where this terminology was used, based on (3), where g ∈ L loc 1 and F : R n → R n is a continuous function. Without any further restrictions on F and g, the state trajectory x(t) may take arbitrary values in R n .…”

“…which, under specific conditions is intimately coupled to the PDS (11), see [6] and also [8], [12], in the sense that the solutions to the PDS (11) and the DI (12) are equivalent. In fact, it can be shown that, for each x ∈ K, the least norm element of −F(x)−g−N K (x) coincides with K (x, −F(x)−g).…”

“…where F is a vector field, g ∈ L loc 1 a locally integrable external input, K is a closed convex set, and K is a projection operator that prevents the solutions from moving outside the constraint set K (see Section III below for a precise definition). These systems are used for studying the behaviour of oligopolistic markets, urban transportation networks, traffic networks, international trade, agricultural and energy markets, see [16] for an overview, and more recently also in control and optimisation [6], [11], [12], [14], [20]. The projection operator is used to guarantee satisfaction of state constraints by the corresponding PDS, i.e.,…”

“…Clearly, one approach could be to study the PDS a posteriori, see, e.g., [16,Ch. 3] or the recent papers [11], [21] using, e.g., local analysis of the stability of equilibria or Lyapunov-based approaches. However, this requires the analysis of discontinuous dynamical systems as in (1), which might be hard.…”