From Lagrange Multipliers to Optimal Control and PDE Constraints

Lagrange Multipliers

Martin J. Gander

Archimedes, Bernoulli, Lagrange, Mechanics Archimedes Varignon Pontryagin, Lions: Bernoulli From Lagrange Multipliers to Optimal Bernoulli’s Rule Lever Double Lever Control and PDE Constraints Varignon example Lagrange Multiplier Constraints Martin J. Gander Multiplier Method Optimization [email protected] Optimal Control Hamiltonian Maximum Principle Pontryagin University of Geneva Adjoint PDE Constraint Optimization Joint work with Felix Kwok and Gerhard Wanner Lions Adjoint Conclusion University of Nice, October 2014 Lagrange Archimedes’ Law of the Lever Multipliers Martin J. Gander Archimedes (287-212 BC): “Two bodies are in equilibrium Mechanics if their weights are inversely proportional to their arm length” Archimedes Varignon Bernoulli Bernoulli’s Rule Lever Double Lever Varignon example Lagrange Multiplier Constraints Multiplier Method Optimization Optimal Control Hamiltonian Maximum Principle Pontryagin Adjoint PDE Constraint Optimization Lions Adjoint Conclusion Lagrange Archimedes’ Law of the Lever Multipliers Martin J. Gander Archimedes (287-212 BC): “Two bodies are in equilibrium Mechanics if their weights are inversely proportional to their arm length” Archimedes Varignon Bernoulli Bernoulli’s Rule Lever Double Lever Varignon example Lagrange Multiplier Constraints Multiplier Method Optimization Optimal Control Hamiltonian Beautiful proof of Archimedes: Maximum Principle Pontryagin 1) Axiom: equal weights at equal distances are in equilibrium Adjoint PDE Constraint Optimization Lions Adjoint Conclusion 2) Redistribute weights to reach symmetric conﬁguration Lagrange Multipliers

Martin J. Gander

Mechanics Archimedes Varignon Bernoulli Bernoulli’s Rule Lever Double Lever Varignon example Lagrange Multiplier Constraints Multiplier Method Optimization Optimal Control Hamiltonian Maximum Principle Pontryagin Adjoint PDE Constraint Optimization Lions Adjoint Conclusion Lagrange Pierre Varignon Multipliers Varignon (1654-1722): Nouvelle Mécanique (frontispiece Martin J. Gander “dont le projet fut donnéen M.DC.LXXXVII”) published Mechanics Archimedes posthumously in 1725 Varignon Bernoulli Bernoulli’s Rule Hundreds of results illustrated in 64 plates of figures: Lever Double Lever Varignon example Lagrange Multiplier Constraints Multiplier Method Optimization Optimal Control Hamiltonian Maximum Principle Pontryagin Adjoint PDE Constraint Optimization Lions Adjoint Conclusion Lagrange Pierre Varignon Multipliers Martin J. Gander

(“règle”) and announces it to Mr. le Chev. Renau (Bernard Mechanics Archimedes Renau d’Eli¸cagaray, “le petit Renau”) in a letter, with copy Varignon to Varignon: Bernoulli Bernoulli’s Rule “Votre projet d’une nouvelle mechanique fourmille d’un Lever Double Lever grand nombre d’exemples, dont quelques uns àen juger par Varignon example les figures paroissent assez compliqués; mais je vous deffie de Lagrange Multiplier m’en proposer un àvotre choix, que je ne resolve sur le Constraints Multiplier Method champ et comme en jouant par ma dite règle.” Optimization Optimal Control Hamiltonian “Your project of a new theory of mechanics is Maximum Principle Pontryagin swarming of examples, some of which, judging from Adjoint PDE Constraint the figures, appear to be quite complicated; but I Optimization Lions challenge you to propose any one of them to me, Adjoint and I solve it on the spot with my so-called rule” Conclusion Varignon had difficulties admitting that all his work over decades was declared to be so easy and contested the general truth of this rule. Lagrange Bernoulli’s Rule in Varignon’s Book Multipliers Martin J. Gander

Mechanics Archimedes Varignon Bernoulli Bernoulli’s Rule Lever Double Lever Varignon example Lagrange Multiplier Constraints Multiplier Method Optimization Optimal Control Hamiltonian Maximum Principle “In every equilibrium of arbitrary forces, no matter how they Pontryagin Adjoint are applied, and in which directions they act the ones on the PDE Constraint Optimization others, either indirectly or directly, the sum of the positive Lions Adjoint energies will be equal to the sum of the negative energies Conclusion taken positively.”

Varignon gives however the wrong date 1717 for the letter of Bernoulli, which was later copied by Lagrange. Lagrange Lagrange Explains Bernoulli’s Rule Multipliers Joseph-Louis Lagrange (1736-1813): “M´ecanique Martin J. Gander

analytique” from 1788 Mechanics Archimedes Varignon Bernoulli Bernoulli’s Rule Lever Double Lever Varignon example Lagrange Multiplier Constraints Multiplier Method Optimization Optimal Control Assume the lever is in equilibrium: Archimedes implies Hamiltonian Maximum Principle Pontryagin P b Pa = Qb =⇒ = Adjoint PDE Constraint Q a Optimization Lions Bernoulli’s idea: apply a “virtual velocity” during an Adjoint inﬁnitely small time interval to displace the lever. Then Conclusion P dq a ∝ dp and b ∝ dq =⇒ = − =⇒ Pdp + Qdq = 0 Q dp Lagrange So Forget Archimedes ! Multipliers Applying the virtual velocity principle, Martin J. Gander

Mechanics Archimedes Varignon Bernoulli Bernoulli’s Rule Lever Double Lever Varignon example Lagrange Multiplier Constraints Multiplier Method we get Optimization Optimal Control Pdp + Qdq = 0. Hamiltonian Maximum Principle b Pontryagin Now if dp = dx for an arbitrary dx, then dq = − a dx, and Adjoint PDE Constraint hence Optimization Lions b Adjoint Pdp + Qdq =0 =⇒ P − Q dx = 0 ∀dx Conclusion a P b =⇒ = (Archimedes) Q a Lagrange A More Complicated System Multipliers Martin J. Gander

Mechanics Archimedes Varignon Bernoulli Bernoulli’s Rule Lever Double Lever Varignon example Lagrange Multiplier Constraints Multiplier Method Optimization Applying the rule of Bernoulli, we get Optimal Control Hamiltonian Maximum Principle Pdp + Qdq + Rdr = 0 (∗) Pontryagin Adjoint PDE Constraint b d bd Optimization If dp = dx, then dq = − a dx, and dr = − c dq = ac dx, and Lions Adjoint b bd Conclusion (∗) =⇒ P − Q + R dx = 0 ∀dx a ac Lagrange A System of Varignon Style Multipliers Martin J. Gander

To use Mechanics Archimedes Varignon Pdp +Qdq +Rdr = 0 (∗) Bernoulli Bernoulli’s Rule Lever Double Lever we compute Varignon example Lagrange 2 2 2 Multiplier p = (x −a) + (y −b) + (z −c) Constraints Multiplier Method q Optimization Optimal Control 1 Hamiltonian dp = ·((x−a)dx+(y−b)dy+(z−c)dz) Maximum Principle p Pontryagin Adjoint and similarly q, dq, and r, dr. PDE Constraint Optimization Lions Inserted into (∗) we get for equilibrium Adjoint Xdx + Ydy + Zdz = 0. Conclusion x−a x−f x−l y−b y−g y−m with X = P p + Q q + R r , Y = P p + Q q + R r z−c z−h z−n and Z = P p + Q q + R r . Lagrange Same System with Constraint Multipliers Martin J. Gander

Mechanics Archimedes Motion (dx, dy, dz) Varignon restricted to a surface L = 0: Bernoulli Bernoulli’s Rule ∂L ∂L ∂L Lever dL = dx+ dy+ dz = 0 (1) Double Lever ∂x ∂y ∂z Varignon example Lagrange Multiplier together with the equation Constraints Multiplier Method Optimization Optimal Control Xdx + Ydy + Zdz = 0 (2) Hamiltonian Maximum Principle Pontryagin ∂L Multiply (1) by λ = −Z/ ∂z and add Adjoint PDE Constraint to (2) to obtain Optimization Lions Adjoint ∂L ∂L ∂L X + λ ∂x · dx + Y + λ ∂y · dy = 0 , Z + λ ∂z = 0 . Conclusion This means applying the virtual velocity argument, without constraints, to Xdx + Ydy + Zdz + λdL = 0 “Il n’est pas difficile de prouver par la théorie de l’élimination des équations linéaires...” Lagrange The Multiplier Method Multipliers Physical interpretation of λ by Lagrange: Martin J. Gander ∂L ∂L ∂L Mechanics λ( ∂x , ∂y , ∂z ) represents the force that holds the particle on Archimedes Varignon the surface L = 0 Bernoulli Bernoulli’s Rule Lever ´ Double Lever “Equation générale” for ALL problems of equilibria: Varignon example Lagrange Multiplier Constraints Multiplier Method Optimization Optimal Control ◮ “Méthode très-simple” in Section IV of the first edition Hamiltonian Maximum Principle from 1788. Pontryagin ◮ Adjoint In the second edition from 1811, Lagrange baptizes the PDE Constraint Optimization method Lions Adjoint Conclusion Lagrange Lagrange Multiplier in Optimization Multipliers Lagrange, Théorie de Fonctions Analytiques (1797): “Théorie des Martin J. Gander

fonctions analytiques, contenant les principes du calcul différentiel, Mechanics Archimedes dégagés de toute considération d’infiniment petits, d’évanouissans, de Varignon limites ou de fluxions, et réduits àl’analyse algébrique des quantités Bernoulli Bernoulli’s Rule finies” Lever Double Lever Constrained optimization problem: Varignon example Lagrange Multiplier f (x) −→ max, g(x) = 0, Constraints Multiplier Method n n m Optimization with f : R → R and g : R → R , m < n. Optimal Control Hamiltonian Maximum Principle Elimination of the constraints: Pontryagin Adjoint − Rm Rn m PDE Constraint x = (y, u), y ∈ , u ∈ Optimization Lions g(x) = 0 and implicit function theorem =⇒ y = y(u) Adjoint Conclusion Unconstrained optimization problem:

f (y(u), u) −→ max . Lagrange Lagrange Multiplier in Optimization Multipliers Martin J. Gander Necessary condition for a local maximum of f (y(u), u): Mechanics Archimedes df ∂f ∂y ∂f Varignon = + = 0 Bernoulli du ∂y ∂u ∂u Bernoulli’s Rule Lever ∂y Double Lever Varignon example For ∂u , implicitly diﬀerentiate constraint g(y(u), u) = 0 Lagrange Multiplier ∂f ∂y ∂f Constraints + = 0, n − m equations Multiplier Method ∂y ∂u ∂u Optimization Optimal Control ∂g ∂y ∂g Hamiltonian + = 0, m(n − m) equations Maximum Principle ∂y ∂u ∂u Pontryagin Adjoint g = 0, m equations PDE Constraint Optimization Lions n + m(n − m) equations for the n unknowns in y and u Adjoint ∂y Conclusion combined, and the m(n − m) unknowns in the Jacobian ∂u

∂y =⇒ A very big system because of ∂u . Lagrange Gaussian Elimination of Lagrange Multipliers Martin J. Gander

∂f ∂y ∂f Mechanics + = 0, n − m equations Archimedes ∂y ∂u ∂u Varignon ∂g ∂y ∂g Bernoulli + = 0, m(n − m) equations Bernoulli’s Rule ∂y ∂u ∂u Lever Double Lever g = 0, m equations Varignon example Lagrange −1 ∂f ∂g Multiplier Multiply 2nd equation by λ := − ∂ ∂ and add to 1st: Constraints y y Multiplier Method Optimization ∂f ∂g Optimal Control + λ = 0, n − m equations Hamiltonian Maximum Principle ∂u ∂u Pontryagin ∂f ∂g + λ = 0, m equations Adjoint PDE Constraint ∂y ∂y Optimization Lions g = 0, m equations Adjoint Conclusion for n unknowns in y and u combined, plus m in λ. Lagrange: get this directly by diﬀerentiating L(u, y, λ) := f (y, u)+ λg(y, u) Lagrange Hestenes’ Optimal Control Problem 1950 Multipliers Martin J. Gander

Mechanics Archimedes Varignon Bernoulli Bernoulli’s Rule Lever T Double Lever f (y, u)dt −→ max, Varignon example Lagrange Z0 Multiplier y˙ = g(y, u), Constraints Multiplier Method 0 Optimization y(0) = y , Optimal Control Hamiltonian y(T ) = y . Maximum Principle T Pontryagin Introduce the Lagrangian Adjoint PDE Constraint T T Optimization Lions L(y, u, λ) := f (y, u)dt + λ (g(y, u) − y˙) dt, Adjoint Z0 Z0 Conclusion where all the variables depend on time. Compute the derivatives with respect to the variables y, u, and λ using variational calculus (as Euler did in E420)! Lagrange Variational Derivatives Multipliers Computing a derivative with respect to y of Martin J. Gander

T T Mechanics Archimedes L(y, u, λ) := f (y, u)dt + λ (g(y, u) − y˙) dt, Varignon Z0 Z0 Bernoulli Bernoulli’s Rule Lever we obtain Double Lever Varignon example d T ∂f T ∂g Lagrange ε dε L(y + εz, u, λ)| =0= 0 ∂y zdt + 0 λ ∂y z − z˙ dt Multiplier R T ∂f ∂Rg T Constraints = + λ + λ˙ z − λ · z| = 0 Multiplier Method 0 ∂y ∂y 0 Optimization Optimal Control R Hamiltonian Maximum Principle Since z(0) = z(T ) = 0, but z(t) arbitrary, we get (and with Pontryagin similar derivatives w.r.t. u and λ) Adjoint PDE Constraint Optimization ∂f Lions 0 = + λB Adjoint ∂u Conclusion 0 y˙ = Ay + Bu y(0) = y , y(T )= yT ∂f T −λ˙ T = + AT λT ∂y Lagrange Hamiltonian Structure Multipliers Defining the Hamiltonian function Martin J. Gander H(y, u, λ) := f (y, u)+ λg(y, u), Mechanics Archimedes Varignon we see that the first order optimality system is Bernoulli Bernoulli’s Rule ∂H Lever y˙ = = g(y, u), Double Lever ∂λ Varignon example ∂H ∂f ∂g λ˙ = − = − − λ , Lagrange ∂y ∂y ∂y Multiplier Constraints Multiplier Method a Hamiltonian system, H is preserved along optimal Optimization Optimal Control trajectories! Hamiltonian Maximum Principle This was already discovered by Carathéodory 1926: Pontryagin Adjoint PDE Constraint Optimization Lions Adjoint Conclusion Lagrange Hamiltonian Structure Multipliers Defining the Hamiltonian function Martin J. Gander H(y, u, λ) := f (y, u)+ λg(y, u), Mechanics Archimedes Varignon we see that the first order optimality system is Bernoulli Bernoulli’s Rule ∂H Lever y˙ = = g(y, u), Double Lever ∂λ Varignon example ∂H ∂f ∂g λ˙ = − = − − λ , Lagrange ∂y ∂y ∂y Multiplier Constraints Multiplier Method a Hamiltonian system, H is preserved along optimal Optimization Optimal Control trajectories! Hamiltonian Maximum Principle Note: The derivative of H w.r.t u is Pontryagin Adjoint PDE Constraint ∂f ∂g Optimization + λ = 0 Lions ∂u ∂u Adjoint Conclusion like the derivative of the Lagrangian L(y, u, λ)! Lagrange Necessary Condition from the Hamiltonian Multipliers Maximizing the Lagrangian Martin J. Gander

T T Mechanics Archimedes L(y, u, λ)= f (y, u)dt + λ (g(y, u) − y˙) dt Varignon Z0 Z0 Bernoulli Bernoulli’s Rule gives along an optimal trajectory y˙ = g(y, u) Lever Double Lever T Varignon example f (y, u)dt −→ max just the original maximization! Lagrange Z0 Multiplier Constraints Multiplier Method Maximizing the Hamiltonian however, Optimization Optimal Control Hamiltonian H(y, u, λ) := f (y, u)+ λg(y, u), Maximum Principle Pontryagin pointwise for each t ∈ [0, T ] (Hestenes’ RAND report 1950) Adjoint PDE Constraint Optimization H(y, u, λ) −→ max with respect to u(t), Lions Adjoint ∂H y˙ = Conclusion ∂λ ∂H λ˙ = − . ∂y Lagrange Discovery of Pontryagin Multipliers Boltyanski, Gamkrelidze, and Pontryagin (1956): On Martin J. Gander

the theory of optimal processes (in Russian) Mechanics Archimedes Varignon Bernoulli Bernoulli’s Rule Lever Double Lever Varignon example Lagrange Multiplier Constraints Multiplier Method Optimization Optimal Control Hamiltonian Maximum Principle Pontryagin Adjoint PDE Constraint Optimization Lions Adjoint Conclusion Lagrange Discovery of Pontryagin Multipliers Boltyanski, Gamkrelidze, and Pontryagin (1956): On Martin J. Gander

the theory of optimal processes (in Russian) Mechanics ◮ Archimedes Controls are often constraint, e.g. −1 ≤ ui ≤ 1 Varignon Bernoulli Bernoulli’s Rule Lever Double Lever Varignon example Lagrange Multiplier Constraints Multiplier Method Optimization Optimal Control Hamiltonian Maximum Principle Pontryagin Adjoint PDE Constraint Optimization Lions Adjoint Conclusion Lagrange Discovery of Pontryagin Multipliers Boltyanski, Gamkrelidze, and Pontryagin (1956): On Martin J. Gander

the theory of optimal processes (in Russian) Mechanics ◮ Archimedes Controls are often constraint, e.g. −1 ≤ ui ≤ 1 Varignon ◮ Solution often on the boundary (“bang-bang”) Bernoulli ◮ y¨ = ±M (Feldbaum 1949) Bernoulli’s Rule Lever ◮ y¨ = ±u, |u|≤ M (Feldbaum 1953) Double Lever Varignon example ◮ Pontryagin discovers the Hamiltonian formulation Lagrange without Lagrange multipliers Multiplier Constraints Multiplier Method Optimization Optimal Control Hamiltonian Maximum Principle (Gamkrelidze (1999) “Discovery of the Maximum Principle”) Pontryagin Adjoint PDE Constraint Optimization Lions Adjoint Conclusion Lagrange Historical Discovery of the Maxmimum Principle Multipliers Martin J. Gander Boltyanski, Gamkrelidze and Pontryagin (1956): Mechanics “This fact appears in many cases as a general principle, which we Archimedes Varignon call the maximum principle” Bernoulli Bernoulli’s Rule Lever Double Lever Varignon example Lagrange Multiplier Constraints Multiplier Method Optimization Optimal Control Hamiltonian Maximum Principle Pontryagin Adjoint PDE Constraint Optimization Lions Adjoint Conclusion Pontryagin (1959): “ In the case that Ω is an open set [...], the variational problem formulated here turns out to be a special case of the problem of Lagrange.” Lagrange Early Work with PDE Constraints Multipliers Martin J. Gander Egorov (1962,1963): Some problems in the theory of optimal control, Optimal control in Banach spaces Mechanics Archimedes Varignon Minimum time control problem for the parabolic equation Bernoulli Bernoulli’s Rule ∂y Lever ∂t + Ay + b(u)y = f + u on Ω × (0, T ) Double Lever y =0 on ∂Ω × (0, T ) Varignon example Lagrange Multiplier Constraints with initial condition y(t0; u)= y0 and target y(t1; u)= yT . Multiplier Method Optimization Optimal Control Hamiltonian Maximum Principle Pontryagin Adjoint PDE Constraint Optimization Lions Adjoint Conclusion Lagrange Early Work with PDE Constraints Multipliers Martin J. Gander Egorov (1962,1963): Some problems in the theory of optimal control, Optimal control in Banach spaces Mechanics Archimedes Varignon Minimum time control problem for the parabolic equation Bernoulli Bernoulli’s Rule ∂y Lever ∂t + Ay + b(u)y = f + u on Ω × (0, T ) Double Lever y =0 on ∂Ω × (0, T ) Varignon example Lagrange Multiplier Constraints with initial condition y(t0; u)= y0 and target y(t1; u)= yT . Multiplier Method Optimization J.-L. Lions: “Le travail de Yu. V. Egorov contient une étude Optimal Control Hamiltonian détaillée de ce problème, mais nous n’avons pas pu comprendre Maximum Principle Pontryagin tous les points des démonstrations de cet auteur, les résultats Adjoint étant très probablement tous corrects.” PDE Constraint Optimization Lions Adjoint Conclusion Lagrange Early Work with PDE Constraints Multipliers Martin J. Gander Egorov (1962,1963): Some problems in the theory of optimal control, Optimal control in Banach spaces Mechanics Archimedes Varignon Minimum time control problem for the parabolic equation Bernoulli Bernoulli’s Rule ∂y Lever ∂t + Ay + b(u)y = f + u on Ω × (0, T ) Double Lever y =0 on ∂Ω × (0, T ) Varignon example Lagrange Multiplier Constraints with initial condition y(t0; u)= y0 and target y(t1; u)= yT . Multiplier Method Optimization J.-L. Lions: “Le travail de Yu. V. Egorov contient une étude Optimal Control Hamiltonian détaillée de ce problème, mais nous n’avons pas pu comprendre Maximum Principle Pontryagin tous les points des démonstrations de cet auteur, les résultats Adjoint étant très probablement tous corrects.” PDE Constraint Optimization Lions Fattorini (1964): Time-optimal control of solutions of Adjoint operational differential equations (proof of the “bang-bang” Conclusion property, no maximum principle) Friedman (1967): Optimal control for parabolic equations Lagrange Jacques-Louis Lions Multipliers Martin J. Gander R. M. Temam (SIAM News, July 10, 2001): Mechanics “A new adventure began for Lions in the early Archimedes Varignon 1960s, when he met (in spirit) another of his Bernoulli Bernoulli’s Rule intellectual mentors, John von Neumann. By then, Lever Double Lever using computers built from his early designs, von Varignon example Neumann was developing numerical methods for Lagrange Multiplier the solution of PDEs from fluid mechanics and Constraints Multiplier Method meteorology. At a time when the French Optimization Optimal Control mathematical school was almost exclusively Hamiltonian Maximum Principle engaged in the development of the Bourbaki Pontryagin program, Lions — virtually alone in France — Adjoint PDE Constraint dreamed of an important future for mathematics in Optimization Lions these new directions; he threw himself into this Adjoint new work, while still continuing to produce Conclusion high-level theoretical work on PDEs.” Lagrange Adjoint Without Lagrange Multipliers Multipliers Lions (1968): Contrôle optimal de systèmes gouvernés par Martin J. Gander

des équations aux dérivées partielles Mechanics Archimedes Varignon Bernoulli 2 J(u)= kCy(u) − zd kH + (Nu, u)U , N self-adjoint, ≥ 0 Bernoulli’s Rule Lever Double Lever Target zd ∈ H, state variable y = y(u) ∈ V , u ∈ Uad , a Varignon example Lagrange closed convex subset U, and PDE constraint Multiplier Constraints ′ Multiplier Method Ay = f + Bu, A : V → V Optimization Optimal Control Hamiltonian J(v) − J(u) ≥ 0 for all v ∈ Uad implies after a short Maximum Principle Pontryagin calculation Adjoint PDE Constraint Optimization (Cy(u) − zd , C(y(v) − y(u)))H + (Nu, v − u)U ≥ 0 Lions Adjoint which is equivalent to Conclusion ∗ (C Λ(Cy(u) − zd ), y(v) − y(u))V + (Nu, v − u)U ≥ 0 Λ : H → H′ canonical isomorphism from H to its dual H′ Lagrange A Clever Guess Multipliers Martin J. Gander ∗ ∗ Defining p(v) ∈ V by A p(v)= C Λ(Cy(v) − zd ), we get Mechanics Archimedes ∗ Varignon (C Λ(Cy(u) − zd ), y(v) − y(u))V + (Nu, v − u)U Bernoulli ∗ = (A p(u), y(v) − y(u))V + (Nu, v − u)U Bernoulli’s Rule Lever Double Lever = (p(u), A(y(v) − y(u))V + (Nu, v − u)U Varignon example = (p(u), B(v − u)) + (Nu, v − u) Lagrange V U Multiplier −1 ∗ Constraints = (ΛU B p(u)+ Nu, v − u)U ≥ 0 Multiplier Method Optimization ∗ ′ ′ Optimal Control where B : V → U is the adjoint of B,ΛU : U → U is the Hamiltonian ′ Maximum Principle canonical isomorphism from U to U . Pontryagin Adjoint Hence the adjoint p(v) permits elimination of y(u), and PDE Constraint Optimization Lions −1 ∗ −1 ∗ Adjoint (ΛU B p(u)+ Nu, u)U = inf (ΛU B p(u)+ Nu, v)U v∈Uad Conclusion

Lions (1968): “La formulation peut être considérée comme un analogue du principe du maximum de Pontryagin” Lagrange Conclusions Multipliers Martin J. Gander We have seen four main ideas: Mechanics Archimedes Varignon 1. Mechanical systems in equilibrium can easily be Bernoulli Bernoulli’s Rule analyzed using “virtual velocities” Lever Double Lever 2. The Lagrange multiplier is just a multiplier from Varignon example Lagrange Gaussian elimination Multiplier Constraints Multiplier Method 3. Using Lagrange multipliers, one can find the adjoint Optimization Optimal Control equation in optimal control Hamiltonian Maximum Principle Pontryagin 4. One can also find the maximum principle of Pontryagin, Adjoint PDE Constraint noticing that the optimality system is Hamiltonian Optimization Lions Adjoint “Constrained Optimization: from Lagrangian Mechanics to Conclusion Optimal Control and PDE Constraints”, Gander, Kwok, Wanner, 2014.