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# Borzí Codes

## Description

Alfio Borzí has released two public domain, single processor Fortran 77 codes.

• Controlla is a full multigrid method for the solution of an optimality system arising from optimal control of the solid fuel ignition model.
• Santafe is for open-loop optimal control of time-dependent reaction-diffusion systems. The code is designed to solve for tracking and terminal observation problems also in combination with receding horizon techniques.

Optimal control (OPC) problems are receiving increasing attention in the scientific community. As an example of optimal control formulation consider the minimization of the following cost functional

J = |U - Z|2/2 + a*|F|2/2
where U satisfies
- U = F
in a given domain Omega.

A way to solve this OPC problem is to reformulate it as an unconstrained minimization problem by introducing the Lagrangian

L = J + < -DELTA U - F, P >
where P is a Lagrangian multiplier. Derive the necessary conditions for a minimum to obtain:
-DELTA U = F
-DELTA P = - ( U - Z )
a*F - P = 0
This is called the optimality system. For a partial list of references concerning the use of multigrid for solving optimality systems see .

The code CONTROLLA.FOR solves the optimality system of the following OPC problem

min ( |U - Z|2/2 + B*|EXP(U)-EXP(Z)|2/2 + a*|F|2/2 )
-DELTA U - D*EXP(U) = F
This reaction-diffusion model describes explosive chemical reaction. Control is required to drive combustion in a desired way or to avoid explosion.

A detailed discussion of this problem and of its multigrid solution is given in . In the linear case (B=0,D=0) CONTROLLA.FOR solves the optimality system given above. Convergence results in the framework of local mode analysis and in the framework of the theory given in  and are presented in . This code has been developed based on techniques and software given in [4, 5].

References
1. Alfio Borzi and Karl Kunisch, The Numerical Solution of the Steady State Solid Fuel Ignition Model and Its Optimal Control, SIAM J. Sci. Comp., 22 (1) (2000), pp. 263-284.
2. A. Borzi, K. Kunisch, and Do Y. Kwak, Accuracy and Convergence Properties of the Finite Difference Multigrid Solution of an Optimal Control Optimality System, Univ. of Graz & Tech. Univ., SFB F003, Rep. No. 245, July 2002, Graz. To appear in SIAM J. Control Optim.
3. J.H. Bramble, Do Y. Kwak, and J.E. Pasciak, Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems, SIAM J. Numer. Anal. 31 (1994), pp. 1746-1763.
4. Achi Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comp., 31 (1977), pp. 333-390.
5. Achi Brandt, Multi-level adaptive techniques (MLAT) for partial differential equations: Ideas and software, Mathematical Software III, J.R. Rice (ed.), Academic Press, New York, 1977, pp. 277-318.

## Archive Contents

There are 6 files in the archives borzi.tgz and borzi.zip:

```borzi:
total 184
-rw-r--r--    339 Sep  4  2006 README.txt
-rw-r--r--  24119 Sep  4  2006 controlla.for
-rw-r--r--   2831 Sep  4  2006 controlla.txt
-rw-r--r--    204 Sep  4  2006 santafe.dat
-rw-r--r--  49197 Sep  4  2006 santafe.for
-rw-r--r--    859 Sep  4  2006 santafe.txt
```

Cheers,
Craig C. Douglas