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The class of admissible controls U is the set of bounded measurable mappings u : [0,T(u)] -i 1. ) E U and xo E R" be fixed. )) such that x(0) = xo. 19) almost everywhere. 20) 0 where f ° is a C'-mapping. , x(0), u) on [0, T], X(0) E M0 and x(T) E M1, then C(u*) < C(u). 2 The Augmented System The following remark is straightforward but is geometrically very important to understand the maximum principle. Let us consider f = (f, fo) and the corresponding system on R"+' defined by the equations I = f (±(t), u(t)), i.
Defined on [to,t1J such that h(to) = h(ti) = 0. ). 2) is satisfied. 2 Hamiltonian Equations The Hamiltonian formalism, which is the natural formalism to deal with the maximum principle, appears in the classical calculus of variations via the Legendre transformation. Definition 8. x - L(t,x,P). 4) 30 2 Optimal Control for Nonlinear Systems Proposition 10. 1) takes the form t, 8L (_. 3 Hamilton-Jacobi-Bellman Equation Definition 9. A solution of the Euler-Lagrange equation is called an extremal.
0 Remark 3. 1. The resolution of c:i = dS on the domain where S : (t, x) F-+ R is a smooth function, is equivalent to solve the Hamilton-Jacobi equation. 2. e. all the extremals are starting at to - E from a single point xo, for e > 0 small enough. Corollary 6. ) be the reference extremal with extremities (to,xo), (t1,x1) and let rd be any curve of T with same extremities. We define by E the excess Weierstrass mapping: E(t, x, x, w) = L(t, x, w) - L(t, x, z) - (w - z) 8x (t, x, z) z = u(t, x), (t, x) E T.