W. Hager. Runge-kutta methods in optimal control and the transformed adjoint system. Numer. Math., 87:247-282, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... to get correct discrete gradient information, two major issues have to be taken into account: Finding a suitable integration scheme for the continous adjoint equation (7) 8) and assembling the gradient as suggested by (9) and (10) The rst part of this question has been addressed in Hager [12] for a general Runge Kutta integration scheme. In contrast to our presentation, Hager pursues a full discretization approach for the underlying control problem, treating the integration scheme as additional constraints. Moreover, he introduces additional optimization variables, corresponding to ....

....calculation gradient information plays an important role for the overall run time. Hence, all savings that can be achieved calculating derivatives have a direct and important impact on the total run time. Therefore, other ways to compute the gradient come into play. As can be seen also from Hager [12], nding the correct integration scheme for the continuous adjoint equation is not a trivial task. We suggest to use Automatic Di erentiation tools to relieve the user of this burden, allowing to conveniently compute the objective gradient. It is interesting to note that Automatic Di erentiation ....

Hager, W.: Runge-Kutta Methods in Optimal Control and the Transformed Adjoint System, Numer. Math. 87, 2000, 247-282.

....finite difference quotients. When working with the OD approach, which has the advantage that the source code of the CNS solver is not required, the discretization of state equation, adjoint equation, and objective function have to be compatible (in a sense not discussed here, see, e.g. [34, 74]) to obtain gradients that are good approximations (i) of the infinitedimensional gradients, and (ii) of the exact discrete gradients. Hereby, requirement (ii) is important for a successful solution of the discrete control problem, whereas (i) crucially influences the quality of the computed ....

....optimal control, measured in terms of the infinite dimensional control problem. This second issue also applies to the DO approach, but for DO it is only important to use compatible discretizations for state equation and objective function. With respect to this interesting topic, we have used [74] as a guideline, to which we refer for further reference. 9.4 Semismooth BFGS Newton Method The implementation of the semismooth Newton method uses BFGS approximations of the Hessian matrix. The resulting semismooth Newton systems have a similar structure as those arising in the step computation ....

W. W. Hager, Runge-Kutta methods' in optimal control and the transformed adjoint system, Numer. Math., 87 (2000), pp. 247-282.

....well known from the optimal control of ordinary differential equations. However there are a few subtle issues that concern the computation of the gradients for the fully discrete problem and the computation of the controls when higher order Runge Kutta time stepping schemes are used. We refer to [20] for a general treatment and to [9] were a discussion of these issues for the semidiscrete problem (18) or (32) 36) can be found. 6. Optimization Algorithm Our numerical results are produced using a nonlinear conjugate gradient algorithm for the solution of the fully discretized problem. Our ....

W. W. Hager. Runge-Kutta methods in optimal control and the transformed adjoint system. Numerische Mathematik, 87:247--282, 2000.

....t 1, are the controls. We point out that the discrete controls h ni can, and often are poor approximations of the control h at t ni . Actually, for some Runge Kutta methods such as the classical Runge Kutta method, t ni = t nj for some i #= j, but in general one has h ni # = h nj . In [25], Hager shows how to compute an approximation hn of the control h at t n . The discrete controls h ni should be viewed as artificial variables needed to obtain good approximations of the 20 S.S. COLLIS, K. GHAYOUR, M. HEINKENSCHLOSS, M. ULBRICH, AND S. ULBRICH states and good approximations of ....

....solve (82) explicitly. In this case the appropriate approximation of the optimal control g # at t n is g # n = 1 # 2 # # n L 1 2 N g (u # n ,g n ,t n) T # # n , 84) n =0, N t 1,whereu # n =u(q # n ) Note that because of our linearity assumption, N g (u,g,t)does not depend on g. In [25] it is proven that under suitable assumptions on (45) 46) the errors max 0#n#N t #h # n h # (t n )#, max 0#n#N t #z # n z # (t n )#, max 0#n#N t ## # n # # (t n )# are of the same order. The order of approximation depends on the order of the Runge Kutta 22 S.S. COLLIS, K. ....

[Article contains additional citation context not shown here]

W. W. Hager. Runge-Kutta methods in optimal control and the transformed adjoint system. Numerische Mathematik, 87:247--282, 2000.

....control problem obtaining an error estimate of order O(h) in the L norm, where h is the size of the uniform mesh. In [18] we analyzed nonlinear optimal control problems with control constraints, obtaining an O(h) estimate in L for the error in the Euler discretization. Most recently, in [29] the convergence rate is determined for general Runge Kutta discretizations of control constrained optimal control problems. These conditions on the coefficients in the Runge Kutta scheme determine whether the discrete (approximating) solution is second , third , or fourth order accurate. In [29] ....

....[29] the convergence rate is determined for general Runge Kutta discretizations of control constrained optimal control problems. These conditions on the coefficients in the Runge Kutta scheme determine whether the discrete (approximating) solution is second , third , or fourth order accurate. In [29] it is assumed that the coefficients in the final stage of the RungeKutta scheme are all positive, while in [21] this positivity requirement is removed for second order Runge Kutta schemes by imposing additional conditions on the coefficients. In [17] Dontchev obtained an estimate for the ....

W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system, Department of Mathematics, University of Florida, Gainesville, FL 32611, January 4, 1999 (http://www.math.ufl.edu/~hager/papers/rk.ps).

....In [52] this result is extended to systems that are nonlinear with respect to the state variable. In [39] O(h ) and O(h) error estimates are obtained for the optimal cost in Runge Kutta discretizations of control systems with discontinuous right hand side. We also point out a companion paper [34] in which conditions are derived for the coe#cients of a Runge Kutta integration scheme that ensure a given order of accuracy in optimal control for orders up to four. The paper [34] focuses on Runge Kutta schemes whose coe#cients in the last stage are all positive, while here this positivity ....

....discretizations of control systems with discontinuous right hand side. We also point out a companion paper [34] in which conditions are derived for the coe#cients of a Runge Kutta integration scheme that ensure a given order of accuracy in optimal control for orders up to four. The paper [34] focuses on Runge Kutta schemes whose coe#cients in the last stage are all positive, while here this positivity condition is removed by working in reduced dimension control spaces. In fact, we show that any second order Runge Kutta scheme for di#erential equations yields a second order ....

[Article contains additional citation context not shown here]

W. W. Hager, Runge--Kutta methods in optimal control and the transformed adjoint system, Numer. Math., to appear.

....the continuous solution in these discrete rst order conditions and estimate a residual. Note though that the discrete rst order conditions seem to have no connection to the continuous rst order condition, the Pontryagin minimum principle. However, we rst showed in [12] and more recently in [9], that when b j 6= 0 for each j, the rst order conditions make more sense (and are more useful) when reformulated in terms of the variables j de ned by j = k 1 s X i=1 a ij b j i ; 1 j s: 19) With this de nition, 16) and (17) are equivalent to the following scheme: k 1 ....

....appear on a summation sign, then the summation is over each index, taking values from 1 to s. Notice that the order conditions of Table 1 are not the usual order conditions [1, p. 170] associated with a Runge Kutta discretization of a di erential equation. The conditions of Table 1 were gotten in [9] by checking the tree based order conditions in [1] However, it was pointed out by Peter Rentrop at the June 4 10, 2000, conference in Oberwolfach, Germany, that these conditions should also follow from the general theory developed for partitioned Runge Kutta methods (see [14, II.15] 15] In ....

[Article contains additional citation context not shown here]

W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system, Numerische Mathematik, 2000, 36 pages.

....to a function of the form ch q , we obtain q 3:09 for (D1) and q 2:02 for (D2) As these examples illustrate, when an optimal control problem is discretized using a Runge Kutta scheme, the order conditions for the optimal control problem and for the di erential equation are not the same. In [18] we show that when b i 0 for each i and when a smoothness and a coercivity condition hold, then the order conditions for 4 Order Conditions (c i = P s j=1 a ij ; d j = P s i=1 b i a ij ) 1 P b i = 1 2 P d i = 1 2 3 P c i d i = 1 6 ; P b i c 2 i = 1 3 ; P d 2 i =b i = 1 ....

.... (x k ; u k ) x 0 = a; 18) 0 k = r x H h (x k ; k 1 ; u k ) N = rC(xN ) 19) r u i H h (x k ; k 1 ; u k ) 0; 1 i s; 20) where k 2 R n , 0 k N 1, and u k 2 R ms is the entire discrete control vector at time level k: u k = u k1 ; u k2 ; u ks ) 2 R ms : In [18] we prove the following theorem: Theorem 1. If Smoothness and Coercivity hold, b i 0 for each i, and the RungeKutta scheme is of order for optimal control, then for all suciently small h, there exists a strict local minimizer (x h ; u h ) of the discrete optimal control problem (4) and an ....

[Article contains additional citation context not shown here]

W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system, Department of Mathematics, University of Florida, Gainesville, FL January 4, 1999 (http://www.math.u .edu/ hager/papers/rk.ps).

No context found.

W. Hager. Runge-kutta methods in optimal control and the transformed adjoint system. Numer. Math., 87:247-282, 2000.

No context found.

W. W. Hager. Runge-Kutta methods in optimal control and the transformed adjoint system. Numerische Mathematik, 87:247--282, 2000.

No context found.

Hager, W. W., "Runge-Kutta methods in optimal control and the transformed adjoint system," Numerische Math-

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC