By Julio Antonio Loría Perez, Françoise Lamnabhi-Lagarrigue, Elena Viatcheslavovna Panteley

This publication comprises chosen contributions via academics on the 3rd annual Formation d’Automatique de Paris. It presents a well-integrated synthesis of the newest pondering in nonlinear optimum keep watch over, observer layout, balance research and structural houses of linear platforms, with no the necessity for an exhaustive literature evaluation. The the world over recognized individuals to this quantity symbolize a number of the so much respected keep an eye on facilities in Europe.

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Additional info for Advanced Topics in Control Systems Theory: Lecture Notes from FAP 2005

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When the Hamiltonian is a quadratic form 1 H(t, z) = tzS(t)z 2 with S(t) symmetric, we get a linear Hamiltonian system z˙ = JS(t)z = A(t)z where A(t) is a Hamiltonian matrix element of sp(n, R). In order to make our geometric analysis, an important issue is the action of the group of symplectic transformations on Hamiltonian vector fields. Let z˙ = J z H(t, z) be a Hamiltonian vector field and consider a change of variables z → ξ = Φ(t, z). The transformation is symplectic if ∂Φ(t, z)/∂z belongs to the symplectic group.

We now strengthen our optimality results to get C 0 -sufficient optimality conditions. Central field, Hamilton-Jacobi-Bellman equation and C 0 -sufficient optimality conditions. Let (¯ z, u ¯) be an extremal defined on [0, T ] of the time-optimal control problem of x˙ = f (x, u), x in a manifold M , extremities being fixed. As previously, we make the following regularity assumptions: (i) The strict Legendre-Clebsch conditions holds along the extremal, ∂2H (¯ z (t), u ¯(t)) < 0, t ∈ [0, T ]. ∂u2 (ii) On each subinterval 0 < t0 < t1 ≤ T the singularity is of codimension one.

16). Decomposing Φ(t) into n × n blocks Φ(t) = Φ1 (t) Φ3 (t) Φ2 (t) Φ4 (t) we define the one parameter family of Lagrangian subspaces L(t) = Φ3 (t) . Φ4 (t) The projection Π : (x, p) → x restricted to L is regular if and only if the matrix Φ3 (t) is invertible. We have Φ˙ 3 Φ˙ 4 = A B tB W − tA Φ3 . Φ4 In the regular case, we introduce R(t) = Φ4 (t)Φ−1 3 (t) which satisfies the symmetric Riccati equation R˙ = W − tAR − RA − RB tBR whose solution is symmetric whenever R(0) is symmetric. Symplectic transformation and generating function.

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