Variable costate

The Lagrange multiplier A is also known as the adjoint or costate variable. It is an undetermined function of an independent variable, which is t in the present problem. Both A and the optimal u are determined by the necessary condition for the optimum of J. The subsequent analysis expands the necessary condition, which is terse as such, into a set of workable equations or necessary conditions to be satisfied at the optimum. 

The above equation is known as the Euler—Lagrange equation in honor of Swiss mathematician Leonard Euler (1707-1783) and French mathematician Joseph Luis de Lagrange (1736-1813). The Euler-Lagrange equation is also called the adjoint or costate equation since it defines the adjoint or costate variable A. 

Show that the costate variables in the optimal control problem... 

Integrate costate equations backward using the final conditions, the control function values, and the saved values of the state variables. Save the values of costate variables at each grid point. 

Integrate costate equations backward from <7 = 1 to 0 using the final conditions, the controls n s, and the state variables y s. Save the values... 

A periodicity condition implies that the initial and final values of a state (or costate) variable are equal to a single value. Thus, in a optimal periodic control problem, the set of state as well as costate equations poses a two point boundary value problem. Either successive substitution or the shooting Newton-Raphson method may be used to integrate the periodic state and costate equations. 

Save the values of costate variables at the grid points. 

If the adjoint function satisfies these equations and boundary conditions, Lis a stationary expression, insensitive to small errors in the density, whose numerical value will yield C. Inspection shows that the Lagrangian has a certain symmetry such that, if N satisfies its equation and boundary conditions, then the Lagrangian is stationary to errors in iV (stationary, in fact, to large errors, since /o and M are not functions of the costate variable). In practice, both equations are perturbed by a change in the control variable and simultaneous errors are made in both functions. For small control perturbations, we anticipate small perturbations in the state and costate variables and that the resulting expression is in error in the cost function only through terms involving the product of small errors. We write... 

Although only one adjoint boundary condition is available from the transversality condition, the adjoint equation and switching function are homogeneous in the costate variable, so that the overall normalization is immaterial in determining an optimum condition. We now have the freedom to impose the additional result of the free end time problem that the Hamilton density vanishes. Boundary conditions to secure this result as well as the transversality condition are... 

We conclude that the relationship between dynamic programming and the optimum control theorem has been established on the basis that — dCjSNii the costate variable N. Dynamic programming is seen to be the Hamilton-Jacobi form of the calculus of variations, whereas the optimum control theorem was the Euler-Lagrange form. 

The Hemithiridoidea, which are characterised by subpentagonal to globose, commonly costate, uniplicate shells, bearing variably developed dorsal septum, uncovered septalium, squama/glotta and raduliform to canaliform crura, contain most ordinary-looking rhynchonellides. 

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