ABSTRACT
In this chapter, we consider optimal control problems with a general Lagrangian F(t, y, u) not necessarily linear in both the state variable y and the control variable u. For the time being, we minimize the performance index (13.12) for a prescribed time interval [t 0, t 1] subject only to the nonlinear equation of state (13.6) and the initial condition (13.7). Similar to problems in the calculus of variations, we append the first variation of the state equation to the first variation of the performance index by a vector Lagrange multiplier (or an adjoint variable) λ(t): δ I = δ J + ∫ t 0 t 1 λ T δ [ f ( t , y , u ) − y • ] d t = [ ψ , y ( y ^ ( t 1 ) ) − λ T ( t 1 ) ] δ y ( t 1 ) + ∫ t 0 t 1 [ ( F ^ , u + λ T f , u ) δ u + { F ^ , y + λ T f ^ , y + ( λ T ) • } δ y ] d t , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749821/5a9bd1b0-aa97-4dfe-adb3-8f69a65a6308/content/eq3613.tif"/>
