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1. Brief Introduction Into Nonlinear Control Theory 1.1 LIE EXTENSIONS OF NONLINEAR CONTROL SYSTEMS ANDREY V. SARYCHEV Abstract. We survey some classical geometric control techniques for studying con- trollability of nite- and innite-dimensional nonlinear control systems. ] 1. Brief introduction into nonlinear control theory 1.1. Basic denitions, see [20, 3]. Denition 1.1.1. Dynamical control system, or C∞ or Cω dynamical polysystem, is a family of C∞ (correspondingly Cω) vector elds parameterized by control parameter u: F = {f(·, u)| u ∈ U}; r U is arbitrary subset of R . The value of u changes with time. Typical choice is measurable bounded dependence u(t): u(·) ∈ L∞([0,T ],U). For most of our purposes piecewise-continuous or even piecewise-constant controls will suce. This latter choice results in a class of piecewise smooth trajectories, where each piece is driven by a vector eld f(·, u0) with u0 xed. tf We will denote the corresponding ow by e u0 . The ow corresponding to a piecewise- constant control has form et1f1 ◦ et2f2 ◦ · · · etN fN , where j . For a general not necessarily piecewise-constant control we fj = fuj , u ∈ U obtain a ow generated by the ODE x˙ = Xt(x), where Xt(·) = f(·, u(t)). We will be interested in the controllability issue which is closely related to the notion of attainability and attainable sets. Denition 1.1.2. A point x˜ is attainable from xˆ in time T (corr. in time ≤ T ) for the system x˙ = f(x, u) if for some admissible control u˜(·) the corresponding trajectory, which starts at xˆ at t = 0, attains x˜ at t = T (at some t ≤ T ). A point x˜ is attainable from xˆ if it is attainable from xˆ in some time T ≥ 0. The set of points attainable from xˆ in time T (in time ≤ T ) is called time-T (time-≤ T ) attainable set from xˆ and is denoted by T (resp. ≤T . The set of points attainable from is called attainable set AF (ˆx) AF (ˆx) xˆ from xˆ and is denoted by AF (ˆx). We say that the system is globally controllable (globally Key words and phrases. nonlinear systems, controllability, relaxation, Lie extension,controlled PDE. Lectures given at Trimester on Dynamical and Control Systems, SISSA-ICTP, Trieste, Fall 2003. 1 2 ANDREY V. SARYCHEV controllable in time T or in time ≤ T ) from xˆ if its attainable set AF (ˆx) (attainable set T , or resp. ≤T ) coincide with the whole state space. AF (ˆx) AF (ˆx)) Now we introduce the notions of global controllability. Denition 1.1.3. We say that the system is globally controllable (globally controllable in time or in time ) from if its attainable set (attainable set T , or T ≤ T xˆ AF (ˆx) AF (ˆx) resp. ≤T ) coincide with the whole state space. AF (ˆx) It is convenient to represent the attainable sets as images of some maps related to control system. Denition 1.1.4. Let us x the initial condition for trajectories of the control system. The correspondence between admissible controls - u(·) and the corresponding trajec- tories of the system is established by input/trajectory map (IT -map). If the there is an output y = h(x) attributed to the system then the correspondence between the control u(·) and the output function h(x(t)) is established by nput/output map (IO-map). If the dynamics of the system is restricted to an interval [0,T ], then the map ITT (u(·)) 7→ x(T ) is called end-point map. Remark 1.1.5. Evidently time-T global controllability of the NS system in observed projection is the same as surjectiveness of the end-point map EPT . Another useful notion tightly related with the issue of optimality is the one of local controllability along a reference trajectory. Denition 1.1.6. Consider a reference trajectory x˜(·) of our control system driven by some admissible control u˜()˙. The system is locally controllable along this trajectory in time if the end-point map is locally onto. T E/PT To dene local controllability properly we have to introduce a metric in the space of admissible controls. In the future it will be metric either generated by L∞-norm, or L1-norm or a weaker metric, such as metric of relaxed controls. 1.2. Elements of chronological calculus, see [1, 2, 3]. Chronological calculus is a formalism for representation and asymptotic analysis of solutions of time-variant dier- ential equations. It has been developed by A.A.Agrachev and R.V.Gamkrelidze at the end of 70's. N Let us consider a time-variant dierential equation in R : x˙ = Xt(x). If this vector eld is complete, i.e. all the solutions are dened ∀t ∈ R then one says that the ODE denes a ow Pt,P0 = Id. It will be convenient to introduce the "operator notation" (P ◦ X)(x) = X(P (x)). NONLINEAR CONTROL SYSTEMS VIA LIE EXTENSIONS 3 Then the dierential equation (1) can be written as d P (x) = P ◦ X (x), dt t t t or after suppressing x: d (1) P = P ◦ X ,P = I. dt t t t 0 Denition 1.2.7. The ow dened by the ODE (1) is called right chronological expo- nential and is denoted by −→ R t exp 0 Xτ dτ. Remark 1.2.8. Left chronological exponential ←− R t denotes the ow dened by exp 0 Xτ dτ the equation (d/dt)Qt = Xt ◦ Qt,Q0 = I. The right chronological exponential admits a series expansion. Indeed let us write Volterra integral equation Z t Pt = I + Pτ1 ◦ Xτ1 dτ1 0 and "iterate" it obtaining Z t Z τ1 Pt = I + I + Pτ2 ◦ Xτ2 dτ2 ◦ Xτ1 dτ1 = 0 0 Z t Z τ1 Z t Z τ1 = I + Xτ2 dτ2 ◦ Xτ1 dτ1 + Pτ2 ◦ Xτ2 dτ2 ◦ Xτ1 dτ1. 0 0 0 0 At the end we obtain so-called Volterra expansion for right chronological exponential. Denition 1.2.9. Volterra expansion or Volterra series for the chronological exponen- tial is (see [AG78,ASkv]): ∞ Z t Z t Z τ1 Z τi−1 (2) −→ X exp Xτ dτ I + dτ1 dτ2 ... dτi(Xτi ◦ · · · ◦ Xτ1). 0 i=1 0 0 0 ∞ N We will consider the Whitney topology in the space of functions ϕ(x) ∈ C (R ) N dened by a family of seminorms k · ks,K , where s ≥ 0, K ⊂ R is compact: ∂ϕ kϕ(x)ks,K = sup (x) x ∈ K, |α| ≤ s . ∂xα For a vector eld X one can dene seminorms componentwise but more elegant denition is kXks,K = sup{kXϕks,K |kϕks+1,K = 1}. If a time-variant vector eld Xt is bounded, analytic , then the series (2) converges provided that R t is suciently small ([1]). 0 kXτ kdτ If a time-variant vector eld Xt is bounded, analytic (and admits an analytic contin- uation onto a neighborhood of the time interval [0,T ] in the complex plane C, then the 4 ANDREY V. SARYCHEV series (2) converges provided that R t is suciently small ([1]). The norm 0 kXτ kdτ kXτ k is the norm of the analytic continuation. In C∞-case the Volterra expansion provides asymptotics for the chronological expo- nential Proposition 1.2.10 ([1]). Let −→ R t be the solution of the equation (1) and Pt =exp 0 Xτ dτ is locally integrable: R T . Then Xτ 0 kXτ ks,K dτ ≤ ρs < +∞ R t C2 kXτ ksdτ (3) kPt ◦ ϕks,K ≤ C1e 0 ; m−1 !! X Z t Z τ1 Z τi−1 P − I + dτ dτ ... dτ (X ◦ · · · X ϕ ≤ t 1 2 i τi τ1 0 0 0 i=1 s,K t m R t Z C2 kXτ ksdτ (4) ≤ C1e 0 (1/m!) kXτ ks+m−1dτ kϕks+m,M , 0 where depend on and is -neighborhood of C1,C2 s, K, ρ M ρ K. The formula (3) indicates that there must be continuity of solutions with respect to the right-hand sides evaluated in 1 s norms R t . In fact a much stronger Xt Lt Cx 0 kXτ kskdτ fact holds: there is continuity with respect to a norm of relaxed controls. This will be referred to later. 1.3. Variational formula, see [1, 2, 3]. Assume that we deal with a "perturbed" ODE: d (5) x˙ = (X + Y )(x) or P = P ◦ (X + Y ),P = I. t t dt t t t t 0 We would like to represent the corresponding ow −→ R t as a multiplica- exp 0 (Xτ +Yτ )dτ tive variation of the non-perturbed ow −→ R t , namely as a composition of this exp 0 Xτ dτ latter with a perturbation ow Ct: t t −→ Z −→ Z exp (Xτ + Yτ )dτ =exp Xτ dτ ◦ Rt, or(6) 0 0 t t −→ Z −→ Z (7) exp (Xτ + Yτ )dτ = Lt◦ exp Xτ dτ. 0 0 To derive the equation for the perturbation ow Lt in (7) we dierentiate this equality. To simplify the notation denote −→ R t by and −→ R t by . To exp 0 Xτ dτ Rt exp 0 (Xτ + Yτ )dτ Pt derive the equation for the perturbation ow Lt in (7) we dierentiate the equality Pt = Lt ◦ Rt , obtaining dP/dt = dLt/dt ◦ Rt + Lt ◦ dRt/dt or Pt ◦ (Xt + Yt) = L˙ t ◦ Rt + Lt ◦ Rt ◦ Xt. Substituting the Lt ◦ Rt instead of Pt in the latter formula we conclude Lt ◦ Rt ◦ Yt = L˙ t ◦ Rt, NONLINEAR CONTROL SYSTEMS VIA LIE EXTENSIONS 5 from where we obtain ˙ −1 Lt = Lt ◦ (Rt ◦ Yt ◦ Rt ). If R is a dieomorphism AdR denotes adjoint action over the Lie algebra of vector elds AdR[X, Y ] = [AdRX, AdRY ], or the group of dieomorphisms: AdR(P ◦ Q) = (AdRP ) ◦ (AdR)Q.
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