By William M. McEneaney

The principal concentration of this publication is the regulate of continuous-time/continuous-space nonlinear platforms. utilizing new strategies that hire the max-plus algebra, the writer addresses numerous sessions of nonlinear keep an eye on difficulties, together with nonlinear optimum keep an eye on difficulties and nonlinear robust/H-infinity regulate and estimation difficulties. a number of numerical recommendations are hired, together with a max-plus eigenvector strategy and an technique that avoids the curse-of-dimensionality. The max-plus-based tools tested during this paintings belong to a completely new type of numerical equipment for the answer of nonlinear regulate difficulties and their linked Hamilton–Jacobi–Bellman (HJB) PDEs; those tools will not be such as both of the mainly used finite aspect or attribute techniques. Max-Plus equipment for Nonlinear keep watch over and Estimation should be of curiosity to utilized mathematicians, engineers, and graduate scholars attracted to the regulate of nonlinear structures during the implementation of lately constructed numerical tools.

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A simple, one-dimensional example indicating the lack of uniqueness is 0 = − −xWx + x2 + 18 Wx2 . There are two C∞ solutions with W (0) = 0: √ √ W 1 (x) = (2 − 2)x2 and W 2 (x) = (2 + 2)x2 , and an infinite number of viscosity solutions such as √ if x ≤ 1 (2 − √2)x2 √ W (x) = (2 + 2)x2 − 2 2 if x > 1. 22). As with the finite time-horizon case, there will be two major parts to the proof of the above statement. Here, we will start with what is referred to as a verification theorem — which states that a solution of the PDE must be the value function.

34) This is an upper bound on the size of ε-optimal u which is independent of T (using e−cf T ≤ 1). 2 Viscosity Solutions In the previous section, we concentrated on the relationship between the DPP and the control problem value function for the example problem classes we will concentrate on. As noted earlier, the DPE is obtained by an infinitesimal limit in the DPP, and takes the form of a nonlinear, first-order Hamilton–Jacobi– Bellman PDE (HJB PDE) in these problem classes. In the finite time-horizon problem, it is a time-dependent PDE over (s, T ) × Rn with terminal-time boundary data.

2 below and [88]) 0 ≤ W (x) ≤ cf γ2 − δ 2 |x| 2m2σ ∀ x ∈ Rn . 23) We now indicate the more specific DPP that one can obtain in this context. 10. 13). Let δ > 0 be sufficiently small . 23) holds, and such that with γ 2 = γ 2 − δ one still has the inequality (γ 2 c2f )/(2m2σ αl ) > 1. Then for any ε > 0, for all x ∈ Rn T W (x) = sup U,ε,|x| u∈U0,T where U,ε,|x| U0,T . 25) . δ δ 2m2σ cf Proof. The following proof is adapted from [88]. 21), T W (x) = sup U u∈U0,T 0 l(ξr ) − γ2 |ur |2 dr + W (ξT ) 2 .