Research Statement

Catalin S. Trenchea

Department of Mathematics Phone: (412) 624-5681University of Pittsburgh Fax: (412) 624-8397Tallahassee E-mail: trenchea@pitt.fsu.eduFL 32306-4120 http://www.math.pitt.edu/∼trenchea/

My research interests are in the area of Numerical Analysis, Control Problems, Fluid Dynamics, Magneto-Hydrodynamics (MHD), Mathematical Biology, and Partial Differential Equations.

1 Research accomplishments

My expertise lies in the numerical analysis of semidiscrete and fully-discrete space-time discretizationsof the control problem, convergence and error estimates, and the development of numerical algorithmsfor finding the optimal solutions. Also, I have experience in writing control problem in an abstract infinitedimensional space framework, the use of analysis and control theory for proving existence of optimal solutions(i.e. solutions that minimize the cost functional and satisfy the state equation), and getting necessaryconditions of optimality for the continuous control problem.

My work is on the numerical analysis for the control problem for partial differential equations withdistributed parameters, i.e., the minimization of cost functionals of the type∫ T

0

(g(y) + h(u)) dt,

for the state variables y and controls u satisfying the state equation

y′ +Ay = Bu,

with initial data or periodic in time, where A is a linear or a nonlinear operator in some Hilbert space, g andh are lower semicontinuous convex functions. Note that for the time periodic problems, even in the linearcase, the differentiability of the state equation is not guaranteed, since uniqueness of the solution is lacking.This implies complications when one is trying to deduce the optimality system for the associated controlproblem.

In the case of nonlinear optimal control problems with initial data, when one is discretizing the equationsin time and space, the uniqueness and the differentiability of the solution of the system with respect tothe control are not preserved. We use the Brezzi-Rappaz-Raviart theory to prove convergence and errorestimates. I successfully solved the optimal control problems for the 2-D MHD with initial data [18], 3-Dmodified Navier-Stokes coupled with Maxwell equations [15], and the reaction-diffusion systems modelingpredator-prey interactions with Holling type II functional response [3, 4, 5].

Systems governed by time-periodic differential equations are relatively difficult to consider in the numer-ical setting because they often possess multiple solutions. I successfully solved the optimal control problemsassociated with the time-periodic elliptic equations [23], wave equation [20], the Euler-Bernoulli equation[21], the Boussinesq equations [22], the Navier-Stokes equations [1, 2], and the magnetohydrodynamic equa-tions [10, 11]. I also solved an inverse problem, i.e., an identification of the nonlinearity in the time-periodicnonlinear wave equation using a least squares approach.

• The velocity tracking problem for MHD flows with magnetic field distributed controls. In[18] we studied the optimal control problem associated with the 2-D magneto-hydrodynamic system (the

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Navier-Stokes equation coupled with the Maxwell equation)

∂u

∂t+ (u · ∇)u− 1

Re∆u+∇p− S curlB ×B = ϕ in Ω× (0, T ),

∂B

∂t+

1Rem

curl(curlB)− curl (u×B) = curlψ in Ω× (0, T ),

divu = 0,divB = 0, in Ω× (0, T ),

with the boundary conditions u = 0, B · ν = 0 and curlB × ν = 0 on ∂Ω × (0, T ), and given initial datau(x, 0) = u0(x), B(x, 0) = B0(x) in Ω . Here u and B denotes the velocity and the magnetic field, respec-tively, while Re, Rem, and S are nondimensional constants that characterize the flow: the Reynolds number,the magnetic Reynolds number and the coupling number, respectively. The cost functional is of quadratictype:

J (u,B, p, ϕ, curlψ) =∫ T

0

∫Ω

(α1

2|u− ud|2 +

α2S

2|B −Bd|2 +

β1

2|ϕ|2 +

β2S

2| curlψ|2

)dx dt

where ud is some desired velocity field, Bd some desired magnetic field.We proved existence of an optimal solution, and the Gateaux differentiability of the state system with respectto controls, which allowed us to use the Lagrange multipliers to derive first-order necessary conditions foroptimality in the continuous problem.We formulated the semi-discrete approximation of the control problem, proved existence of optimal solutionsand the convergence for ∆t→ 0 of the solution of the semi-discrete-in-time control problem to the solutionof the continuous control problem. To obtain the necessary conditions for optimality in the semi-discretecase we used a different approach, since throughout discretization we end up with a system of steady MHDequations, known to lack uniqueness. We have defined two nonlinear maps that allowed us to reformulatethe semi-discrete control problem in an equivalent form, for which we proved existence of a nonzero Lagrangemultiplier by showing that the directional derivatives of these maps have closed ranges.For the fully time-space discretization, in order to have stable and accurate approximations, we consideronly conforming finite elements, that satisfy the inf-sup condition. The finite element spaces for the velocityand magnetic fields are the same (Taylor-Hood in the simulations). We have formulated the fully-discreteapproximation of the control problem, stated the existence and convergence of the solution (which can beproved in a similar way as for the semi-discrete case for conforming finite elements). We formulated thefully discrete optimality system, derived a gradient type iterative algorithm and proved its convergence tothe optimal solution. In a numerical simulation we showed that the flow can be driven from a clockwisespinning initial velocity data to a counterclockwise target (Figure 1) using only magnetic field controls(see Figure 2). By tuning the weights in the cost functional we have put more emphasize in matchingthe velocity target than matching the magnetic field target. The constants used in our simulation are:Re = 50, Rem = 1, S = 10, α1 = 100, α2 = 10, β1 = β2 = .01, N = 50,∆t = .005.

Figure 1: Velocity target ud and magnetic field target Bd

• Optimal control of a nonlinear system in ecology. We consider in [5] the mathematical formula-tion and analysis of an optimal control problem for a nonlinear ‘fish-phytoplankton-zooplankton’ reaction-diffusion system (considered in [3, 4]) through the adjustment of f , the rate of removal of zooplankton by

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Figure 2: Controlled velocity u and magnetic field B at time steps N = 0, 3, 4, 5, 6, 7, 9, 16, 25, 50

the fish population: ∂u

∂t= d1∆u+ ru(1− u)− auv

1 + bu, in Ω× (0, T )

∂v

∂t= d2∆v +

auv

1 + bu−mv − f(x, t)

gv2

1 + g2v2, in Ω× (0, T ).

Note that the control enters the state equation in a multiplicative way, the state has to be positive asdensities, the control has to be positive as a rate, and the cost functional minimizes the time derivativeof f , modeling the goal of minimum change in the outside intervention. Existence of optimal solutions isproved and first-order necessary conditions for optimality are derived. A semi-implicit Galerkin finite ele-ment method with piecewise linear continuous basis functions and a variable size-step gradient algorithmwere used for the numerical results. We show the ability to drive the plankton dynamics from a chaoticregime to an arbitrary ordered state (a rotating one-armed Archimedian spiral), see Figures 3, 4.

• Finite element approximations of spatially extended predator-prey interactions with Hollingtype II functional response. In [3] we continued the work from [4] by presenting two fully practical piece-wise linear finite element methods. A priori estimates and error bounds for semi-discrete and fully-discretefinite element approximations, along with numerical results in one and two-space dimensions are presented.

• Global existence reaction diffusion systems modeling predator-prey interactions with Hollingtype II functional response. In [4] we proved that the classical solutions of a well known class of reaction-

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(a) (b)

Figure 3: Uncontrolled (a), optimally controlled (b), for phytoplankton densities u at time T = 100. Parameter values:

d1 = d2 = 0.05, r = 1, a = b = 20, m = 0.8, g = 10, α = 10−5.

(a) (b)

Figure 4: Target (a), and the control f scaled with a factor of 10−5 (b)

diffusion systems in ecology are globally well posed and nonnegative, given any bounded nonnegative initialdata. We have considered predator-prey interactions with logistic growth of the prey in the absence ofpredators, and Holling type II functional response for the predators, given by

∂u

∂t= ∆u+ u(1− u)− vh(au) in Ω× (0, T ),

∂v

∂t= δ∆v + bvh(au)− cv in Ω× (0, T ),

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω,∂u

∂ν=∂v

∂ν= 0, on ∂Ω× (0, T ),

where h is the functional response. We constructed invariant regions in the equal diffusion case which throughsemigroup theory leads to global existence of solutions, while for the distinct diffusion coefficient case weused a different approach based on Lyapunov-type conditions satisfied by the kinetics.

• Analysis and discretization of an optimal control problem for the time-periodic MHD equa-tions. In [10, 11] we have considered the mathematical formulation and the analysis of a time-periodicoptimal control problem associated with the tracking of the velocity and the magnetic field of a viscous,incompressible, electrically conducting time-periodic fluid in a bounded two-dimensional domain through

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the adjustment of distributed controls. Existence of optimal solutions is proved and first-order necessaryconditions for optimality are used to derive an optimality system of partial differential equations whosesolutions provide optimal states and controls. In the time-periodic case, the uniqueness of the solution islacking even for the continuous equation. Hence we have formulated a sequence of penalized optimal controlproblems that converges to our original problem and for which we were able to prove necessary conditionsfor optimality.We have formulated semi-discrete-in-time approximations for our control problem, we proved the existenceof optimal solution and convergence of the solution of the semi-discrete-in-time optimal control problemto the solution of the corresponding continuous optimal control problem when the time-step ∆t → 0. Wederived optimality conditions for the semi-discrete-in-time control problem. For error estimates we usedthe Brezzi-Rappaz-Raviart theory to prove that for each nonsingular branch of solutions for the continuousoptimality system there exists a neighborhood O of the origin and for N = T/∆t big enough, there exists aunique branch of solutions for the semi-discrete-in-time optimality system such that the difference of the twosolutions lies in O, and we obtained error estimates for the velocity and magnetic field in L∞(0, T ;H2(Ω)),pressure in L∞(0, T ;H1(Ω)), of order ∆t.

• Noncooperative optimization of controls for time periodic Navier-Stokes systems with multi-ple solutions. Time-periodic systems governed by differential equations are somewhat difficult to considerin the numerical setting because they may possess many solutions. The number of solutions of such systemsmay be finite or infinite. Further, some trajectories which are exactly time-periodic over a given periodmight only approximately solve the governing equation, whereas nearby trajectories which exactly solve thegoverning equation might only be approximately time-periodic over the given period. The difficulty of thetime-periodic setting is compounded in the case of systems governed by the Navier-Stokes equation, as thesolutions of such systems in the time-evolving setting may be chaotic and multiscale. When consideringthe optimization of controls for such systems in the time-periodic setting, the situation is thus particularlydelicate, as one doesn’t know a priori which time-periodic solution (or approximate solution) one shoulddesign the controls for. In [1, 2], the idea of noncooperative optimization is applied in an attempt to developa tractable framework to solve the problem of optimization of controls for time-periodic Navier-Stokes sys-tems. The noncooperative aspect of the optimization, however, is somewhat nonstandard: the best controlsare found for the worst (of the many) time-periodic solutions of the governing equation, i.e. we compute

infu∈L2(Q)

supv∈L2(Q)

(12

∫Q

|v(x, t)|2dxdt+∫ T

0

h(u(t))dt

)

subject to ∂v∂t

− ν∆v + (v · ∇)v +∇p = Bu + f , in Ω× R,

div v = 0 in Ω× R,v(x, t+ T ) = v(x, t) in Q = Ω× (0, T ),

where u is T -periodic input, f is a T -periodic source field, and B is a linear continuous operator. As thenumber of solutions may be finite, we have employed a technique developed by Barbu (1998) of first lookingat a suitable approximation of the time-periodic system of interest with an infinite number of solutions,finding the solution to this approximate system with a gradient-based algorithm leveraging an adjoint anal-ysis, then refining the level of approximation until we have solved (with a sufficient level of accuracy) theoptimization problem we are actually interested in.

• Periodic optimal control of the Boussineq equation. In [22] I also considered the periodic op-timal control problem

infu,u1,v,θ

∫Q

(|v − v0|2 + |curlv|2 + |θ − θ0|2 + h(u) + |u− u0

1|2)dxdt

5

for an incompressible fluid flow coupled with thermal dynamics in two-space dimensions, the Boussinesqsystem (Navier-Stokes coupled with heat equation) that appears in thermodynamics and combustion:

∂v∂t

− ν∆v + (v · ∇)v − σθ +∇p = m(x)u,

∂θ

∂t− χ∆θ + v · ∇θ = u1,

div v = 0,v(x, t+ T ) = v(x, t), θ(x, t+ T ) = θ(x, t),

with homogeneous boundary conditions. Here v is the velocity vector, θ is the temperature, p stands for thepressure, while u is the velocity control and u1 is the temperature control. The term m(x) allows us to actwith velocity controls only on a portion of the space domain. The objective is to determine u and u1 in sucha way that the velocity vector and the temperature distribution are as close as possible to the desired velocityv0 and the temperature distribution θ0, and the turbulence are minimal. Due to the periodicity in time ofv and θ the solution is not unique, (the number can be infinite for some values of the velocity-temperaturecontrol couple (u, u1)). The minimization is considered on a manifold of solutions (v, θ,u, u1). The lack ofdifferentiability of the mapping control 7→ state is compensated for by the consideration of a sequence ofpenalized approximating problems:

infu,u1,v,θ,Φ,Ψ

∫Ω

(|v − v0|2 + hε(u) + |curlv|2 + |θ − θ0|2 + |u− u0

1|2

+|v − v∗|2 + |θ − θ∗|2 + |u− u∗|2 + |u− u1∗|2 +

1ε|Φ|2 +

1ε|Ψ|2

)dt

over (v, θ,u, u1,Φ,Ψ) subject to

∂v∂t

− ν∆v + (v · ∇)v − σθ +∇p = m(x)u + Φ,

∂θ

∂t− χ∆θ + v · ∇θ = u1 + Ψ,

div v = 0,v(x, t+ T ) = v(x, t), θ(x, t+ T ) = θ(x, t).

Necessary conditions of optimality were proved, which in the limiting case yield the optimality conditionsfor our initial control problem.

• Optimal control of the periodic equation with internal control. In [20] I studied the existenceand the maximum principle for an optimal control problem governed by the time periodic vibrating stringequation

ytt(x, t)− yxx(x, t) = m(x)u(x, t), x ∈ (0, π), t ∈ R,y(x, t+ T ) = y(x, t), x ∈ (0, π), t ∈ R

on (0, π)× (0, T ), with Dirichlet boundary conditions and internal controllers supported on an open subsetω ⊂ (0, π). We used intensively the fact that when the period T is a rational multiple of π, the range R(A)of the one-dimensional wave operator is closed, A−1 ∈ L(R(A), R(A)) and is compact on R(A). The stateconstraint problem is also analyzed.

• Identification for nonlinear periodic wave equation. In [9] we studied an inverse problem, i.e.,the identification of the nonlinearity g for the nonlinear periodic 1-D wave equation

ytt(x, t)− yxx(x, t) + g(y(x, t)) 3 f(x, t),y(x, t+ 2π) = y(x, t),

on Q = (0, π)× (0, 2π), with homogeneous boundary conditions, g in the class of subdifferentials A = g =∂j : j ∈ K, with K = j : R → R, convex, continuous and β1|r|2 + γ1 ≤ j(r) ≤ β2|r|2 + γ2,∀r ∈ R. The

6

identification problem consists in determining, for given y0, a function g ∈ A such that y(g) = y0. Of course,such a g may not exist. The least-squares approach leads us to the minimization problem

min‖y − y0‖2L2(Q) : Wy + g(y) 3 f, y ∈ L2(Q), q = ∂j, j ∈ K

for given y0 ∈ L2(Q). Furthermore we approximate this by a family of penalized minimization problems

min‖y − y0‖2L2(Q) +

1ε

∫Q

(j(y) + j∗(v)− yv) dxdt : y ∈ L2(Q), j ∈ K, v ∈ L2(Q),Wy + g(y) = f

for which we proved existence of solutions and showed that when ε→ 0 these solutions converge to the solu-tion of our original problem. With an iterative gradient type algorithm we compute the numerical solution

Figure 5: y0, the numerical solution y and j(y)

for y, j(y) corresponding to the given solution y0(see Figure 5).

• Periodic optimal control of the Euler-Bernoulli equation. I have studied the optimal controlproblem associated with the motion of the Euler-Bernoulli equation ytt(x, t) + yxxxx(x, t) = m(x)u(x, t) + f(x, t), x ∈ (0, π), t ∈ R,

y(x, t+ T ) = y(x, t), x ∈ (0, π), t ∈ R,y(0, t) = yx(0, t) = 0, yxx(π, t) = yxxx(π, t) = 0, t ∈ R,

on a beam fixed at one end and without torque and lateral force at the free-end point in [21]. I provedthat the operator associated with this equation is self-adjoint, has a closed range and a finite-dimensionalnull-space, and the inverse is compact on the range. This allowed me to derive existence of optimal solutions,the maximum principle for the control problem, and an existence result for the periodic Hamiltonian systemassociated with the operator. We note that, as in the case of the time-periodic wave equation, the controlsare acting only on a subdomain of the physical domain.

• Optimal control of an elliptic equation under periodic conditions. In [23] I studied the followingtwo control problems

Minimize L1(y, u) =∫ T

0

(y(0, t)− g1(t))2dt and L2(y, u) =

∫ T

0

(y(x, 0)− g1(x))2dx

associated with an elliptic equation under periodic condition

ytt(x, t) + (u(x)yx(x, t))x = f(x, t) in Q = (0, 1)× (0, T )yx(0, t) = 0, y(1, t) = 0 in (0, T )y(x, 0) = y(x, T ), yt(x, 0) = yt(x, T ) in (0, 1)

where the set of admissible controls is given by

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Figure 6: The optimal control u∗ and the optimal state y∗

U =u ∈ Lip(0, 1); |u′(x)| ≤ ρ a.e. in (0, 1),0 < a ≤ u(x) ≤ b,∀x ∈ [0, 1], u(0) = α, u(1) = β

with a < α < β < b, ρ ≥ α+ β − 2a > 0 and g1 ∈ C2([0, T ]), g2 ∈ L2(0, 1). The hypotheses on f are

f ∈ C1(Q), f(x, t) > 0, fx(x, t) < 0 in Q.

The equation models a strip heat conductor (0, 1)×R with periodic conditions in time and zero temperatureat x = 1. Using the maximum principle for elliptic equations and the structure of the normal cone NU(u),under some conditions for g1 and g2 we proved that the optimal controls are continuous piecewise functionsof the form

u∗(x) =

α− ρx for x ∈ [0, (α− a)ρ−1],α, for x ∈ [(α− a)ρ−1, (a+ ρ− β)ρ−1]β + ρ(x− 1), for x ∈ [(a+ ρ− β)ρ−1, 1]

,

(see the optimal control and the corresponding optimal state in Figure 6).

• Value function and optimality conditions for a boundary control problem. In [19] we char-acterized the value function associated with a control problem governed by a semilinear parabolic equationwith boundary control by an appropriate Hamilton Jacobi Bellman equation (in the viscosity sense), andderived optimality conditions from the knowledge of the value function.

2 Work in progress and future plans

• Optimal control problem for 3-D coupled modified Navier-Stokes and Maxwell equations.We consider the mathematical formulation and the analysis of an optimal control problem associated withthe tracking of the velocity and the magnetic field of a viscous, incompressible, electrically conducting fluidin a bounded three-dimensional domain through the adjustment of distributed controls.The mathematical description of the control problem proceeds as follows. Let Ω be a bounded domain in R3

with a boundary ∂Ω ⊂ C2. Let v denote the velocity, p the pressure, and H the magnetic field. Denote by fthe velocity control and by curl j the magnetic field control. For given T > 0, the cost functional is definedby:

L(v,H, f , curl j) =∫ T

0

∫Ω

(α1

2|v − vd|2 +

α2

2|H−Hd|2 +

β1

2|f |2 +

β2

2| curl j|2

)dx dt

where vd is some desired velocity field, Hd some desired magnetic field.We wish to minimize L subject to the constraints which are the modified Navier-Stokes equations (formulated

8

by Ladyzhenskaya) coupled with the Maxwell equations:

vt + (v · ∇)v − divΞ(v) + µH× curlH +∇p = f

divv = 0

µHt +1σ

curl(curlH) + µ (v · ∇H−H · ∇v) =1σ

curl j

divH = 0

with v = (v1, v2, v3), H = (H1,H2,H3), and

Ξ(v) =∂D(ε)∂ε

∣∣∣ε=ε(v)

,

where

ε(v) = (εij(v)), εij(v) =12(vi,j + vj,i), vi,j ≡

∂vi

∂xj

supplemented by the initial data:

v|t=0 = v0 and H|t=0 = H0 in Ω

and one of the following sets of boundary conditions

v|ST= 0, Hn|ST

= 0, and (curlH)τ |ST= 0,

where ST = ∂Ω× [0, T ], or

v,H and p are periodic with respect to xk, k = 1, 2, 3.

Here Hn = H · n is the projection of H onto the outer normal n of ∂Ω and uτ is the projection of thevector u onto the vector tangential to ∂Ω. µ > 0 denotes the constant magnetic permeability and σ > 0 theconstant electric conductivity. The global unique solvability of the state equations was proved for the three-dimensional case, under the assumption δ ∈ [1/4, 2]. For the two-dimensional domains Ω, the parameter δcan be any nonnegative number. The potential D(·) is a smooth function having the properties:

1. D : M3×3sym → R1

+ = [0,∞) and D ∈ C2(M3×3sym);

2. ν1m(ε) ≤ D(ε) ≤ ν2m(ε), where m(ε) = |ε|2 + |ε|2+2δ);

3. ν3m(ε) ≤ ∂D(ε)∂εij

εij ≤ ν4m(ε);

4. ν5(1 + |ε|2δ)|κ|2 ≤ ∂2D(ε)∂εij∂εkl

κijκkl ≤ ν6(1 + |ε|2δ)|κ|2;

5. ∂3D(ε)∂εij∂εkl∂εmn

κij`klπmn ≤ ν7|ε|2δ−1|κ||`||π|

with νk > 0, k = 1, 2, . . . , 7, constants and κ, `, π arbitrary elements in M3×3sym. For the Navier-Stokes

equations, we have D(ε) = ν|ε|2 and divΞ(v) = ν∆v.In [15] we showed the existence of optimal solutions, we proved the Gateux differentiability for the MHD

system with respect to controls, and we obtained the optimality system.

• Finite element approximation on Ladyzhenskaya model for MHD equations. In [?] we ex-amine certain analytic and numerical aspects of optimal control problems for a Ladyzhenskaya model forincompressible flows. The controls we consider are of distributed type; the functionals minimized are theL2-distance of candidate flow to some desired flow and the viscous drag on bounding surfaces. The existenceof optimal solutions, the regularity of solutions of this system, and the justification for the use of Lagrangemultiplier techniques to derive a system of partial differential equations from which optimal solutions may

9

be deduced was proved in ([15]). We consider approximations by finite element methods of solutions of theoptimality system and examine their convergence properties.

• Numerical analysis of nutrient-plankton optimal control problem. In [6] we present the nu-merical approximation of the time-dependent optimal control problem of tracking the plankton densitiesfor the nonlinear ‘fish-plankton’ reaction diffusion system considered in [5]. We define and analyze thesemidiscrete-in-time and the full space-time discrete approximations of the optimality system, prove conver-gence of the approximations and error estimates, and show the convergence of the solution of the iterativegradient algorithm to the solution of the fully discrete optimality system. We provide results of computa-tional experiments.

• Robust control of a nonlinear ‘fish-phytoplankton-zooplankton’ optimal control problem.In [7] we generalize the optimal control problem considered in [5, 6] in order to make the effective controlalgorithms insensitive to external disturbances. Our aim is to put such algorithms into a rigorous math-ematical framework, to give the conditions on the initial data, the parameters of the cost functional andthe regularity of the problem such that the existence and uniqueness of the solution of the robust controlproblem can be proven. We also propose an appropriate numerical method.

• Analysis of a regularized spatially extended predator-prey system. Another classical spatiallyextended predator-prey system has the form

dp

dt= d2∆p+Bp(1− p/h),

dh

dt= d1∆h+ h(1− h)−Ahp/(h+ C),

where p(x, t) and h(x, t) are population densities for predators and prey, and d2 and d1 are diffusion coef-ficients respectively. For apropriate choices of the positive parameters A,B, and C, the limit kinetics havea limit cycle. From a rigorous mathematical point of view these equations are challenging because of the‘singular term’. To overcome this difficulty we undertake the analysis and the numerical analysis of theassociated regulariseed problem, where we modify the singular term via

Bp(1− p/h) −→ Bp

(1− p

h+ ε

),

for some arbitrary, but small, ε.

• Time-periodic solutions for the Ginzburg-Landau equations. In [12] we are studying the ex-istence of time-periodic solutions for the Ginzburg-Landau equations in two space dimensions

∂ψ

∂t− i

(k

σJy

)ψ − ψ + |ψ|2ψ +

(i

k∇+ A

)2

= 0 in Ω× R+

σ∂A∂t

+∇×∇×A + |ψ|2A +i

2k(ψ∗∇ψ − ψ∇ψ∗) =

(0J

)in Ω× R+.

with constant applied magnetic field H and current J , with the boundary conditions ∇ψ · n = 0 on ∂Ω× R+,(∇×A)× n = H× n on ∂Ω× R+,A · n = 0 on ∂Ω× R+.

Here ψ denotes the (complex-valued) order parameter and A is the magnetic potential. The numericalresults confirm that with steady forcing, for some values of the parameters, the equation has a time-periodicbehavior.

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• Observability of primitive equations of geophysics. In [14] we consider the observability andcontrollability problem associated with the primitive equations of the large-scale ocean

∂v∂t

+∇vv + w∂v∂z

+1ρ0p+ 2Ω sin θ k× v − µv∆v − νv

∂2v∂z2

= Fv,

∂p

∂z= −ρg,

divv +∂w

∂z= 0,

∂T

∂t+∇vT + w

∂T

∂z− µT ∆T − νT

∂2T

∂z2= FT ,

∂S

∂t+∇vS + w

∂S

∂z− µS∆S − νS

∂2S

∂z2= FS ,

ρ = ρ0 (1− βT (T − Tr) + βS(S − Sr)) ,

where v is the horizontal velocity of the water, w is the vertical velocity, ρ, p, T are the density, pressureand temperature respectively, S is the concentration of the salinity and g is the gravity.

• Computational methods for stabilization of parabolic and Navier-Stokes equations. In [13]we study computational methods for stabilization of parabolic and Navier-Stokes equations defined in a 2-Dbounded domain Ω, near a steady state v by means of feedback controls defined on a part Γ of boundary ∂Ω.

• Analysis and approximation for a linear feedback control for tracking the velocity in MHDflows. In [16] we study some systematic approaches to the mathematical formulation and numerical reso-lution of the linear feedback control problem for tracking the velocity and magnetic field in MHD flows ina bounded two-dimensional domain with bounded distributed controls. Semidiscrete-in-time and full space-time discrete approximations are presented along with computational results.

• The velocity and magnetic field tracking problem for MHD flows with distributed controls.In [17] we present systematic approaches to the mathematical formulation and numerical approximation ofthe time-dependent optimal control problem of tracking the velocity and the magnetic field for MHD flows ina bounded, two-dimensional domain with boundary control. We study the existence of optimal solutions andderive an optimality system from which optimal solutions may be determined. We also define and analyzesemidiscrete-in-time and full space-time discrete approximations of the optimality system and a gradientmethod for the solution of the fully discrete system. We provide results of some computational experiments.

References

[1] T. Bewley and C. Trenchea, Noncooperative optimization of controls for time periodic Navier-Stokessystems with multiple solutions, AIAA 2002-2754.

[2] T. Bewley and C. Trenchea, Noncooperative optimization of controls for time periodic Navier-Stokessystems with multiple solutions, submitted to SIAM J. Control Optimization.

[3] M.R. Garvie and C. Trenchea, Finite element approximations of spatially extended predator-prey inter-actions with Holling type II functional response, submitted to SIAM J. Numer. Anal.

[4] M.R. Garvie and C. Trenchea, Global existence for reaction diffusion systems modelling predator-preyinteractions with the Holling type II functional response, submitted to J. Math. Anal. Appl.

[5] M.R. Garvie and C. Trenchea, Optimal control of a nonlinear ‘fish-phytoplankton-zooplankton’ system,to appear in SICON.

[6] M.R. Garvie and C. Trenchea, Numerical analysis of a nutrient-plankton optimal control problem, inpreparation.

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[7] M.R. Garvie and C. Trenchea, Robust control of a nonlinear ‘fish-phytoplankton-zooplankton’ system,in preparation.

[8] M.R. Garvie and C. Trenchea, Analysis of a regularized spatially extended predator-prey system, inpreparation.

[9] C. Morosanu and C. Trenchea, Identification for nonlinear periodic wave equation, Appl. Math. Opti-mization, 44 (2001), 87–104.

[10] M. Gunzburger and C. Trenchea, Optimal control of time-periodic MHD equations, Nonlinear Anal.,Theory Methods Appl., Proceedings for the Fourth World Congress of Nonlinear Analysis WCNA-2004.

[11] M. Gunzburger and C. Trenchea, Analysis and discretization of an optimal control problem for thetime-periodic MHD equations, J. Math. Anal. Appl. 308 (2005), no. 2, 440–466.

[12] M. Gunzburger and C. Trenchea, Periodic solutions for the Ginzburg-Landau equations, in preparation.

[13] M. Gunzburger and C. Trenchea, Computational methods for stabilization of parabolic and Navier-Stokes equations, in preparation.

[14] M. Gunzburger and C. Trenchea, Controllability and observability of primitive equations of geophysics,in preparation.

[15] M. Gunzburger and C. Trenchea, Analysis of optimal control problem for three-dimensional coupledmodified Navier-Stokes and Maxwell equations, to appear in J. Math. Anal. Appl.

[16] M. Gunzburger, J. Peterson, and C. Trenchea, Analysis and approximation for a linear feedback controlfor tracking the velocity in MHD flows, in preparation.

[17] M. Gunzburger, J. Peterson, and C. Trenchea, The velocity and magnetic field tracking problem forMHD flows with boundary controls, in preparation.

[18] M. Gunzburger, J. Peterson, and C. Trenchea, The velocity and magnetic field tracking problem forMHD flows with distributed controls, submitted.

[19] M. Quincampoix and C. Trenchea, Value function and optimality conditions for a boundary controlproblem, submitted to J. Differ. Equations.

[20] C. Trenchea, Optimal control of the periodic string equation with internal control, J. OptimizationTheory Appl., 101(1999), 429–447.

[21] C. Trenchea, Periodic optimal control of the Euler-Bernoulli equation, Commun. Appl. Anal. 7 (2003),no. 1, 115–125.

[22] C. Trenchea, Periodic optimal control of the Boussinesq equation, Nonlinear Anal., Theory MethodsAppl. 53A, No.1, 81-96 (2003), 81-96.

[23] C. Trenchea, Optimal control of an elliptic equation under periodic conditions, Memoriile SectiilorStiintifice ale Academiei Romane, Ser. IV 25 (2002), 23–35 (2005).

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