José de Jesús Martı́nez
Department of Mathematics
Research Statement
Iowa State University
Ames, IA 50011
Research Statement
My research is in the area of control theory of partial differential equations (PDE). Control theory is concerned with utilizing inputs or “feedbacks” to improve some aspect of the evolution of
a system of dynamical equations. The dynamics are typically described by differential or partial
differential equations, the latter being my main interest. While control theory is central to many
areas of engineering, the theory needed to describe control theoretical properties of PDE involve
advanced mathematical topics such as PDE theory, operator theory, functional analysis, harmonic
analysis along with advanced numerical methods. The foundations of the modern approach toward
control of PDE go back to Richard Kalman in the 1960s when he developed finite dimensional systems theory. Subsequently, mathematicians such as Lions, Russell, Fattorini, Littman, Komornik,
Lagnese, Zuazua, Lasiecka and Triggiani generalized these ideas in the 1970s and 1980s to infinite
dimensional systems where PDE are involved.
In particular, my research has been motivated by the work of Hansen and Zuazua in [?] where
they consider a hybrid system consisting of two vibrating strings connected by a point mass. In this
paper, it is observed that such a system is well posed in an asymmetric space in which solutions
have one degree more of regularity on one side of the point mass. Furthermore, they show that
this is due to a lack of spectral gap by the presence of the point mass. This result is generalized by
Zuazua and Castro in [?] to the case of a beam equation. My research problem consists of studying
the thermal component in addition to the mechanical one. This focus leads to a thermoelastic
system with an interior point mass. The first step is to understand the thermal properties. For
that reason, we consider a system of heat equations coupled by a point mass. Furthermore, we
1
allow for the parameters of specific heat, density, conductivity and length of the rod to be different
to either side of the point mass. The problem can be written as an evolution equation of the form
d
y(t) = Ay(t) + Bf (t),
dt
where B is the control operator acting on the input control f . This operator may correspond to that
of a Dirichlet boundary control or Neumann boundary control. In other words, one may control
either the temperature or flux at one end the rod. The problem of null controllability in time T > 0
is that of finding f ∈ L2 (0, T ) such that given any initial state y0 in a Hilbert space H, we can
reach the zero state in time T , i.e. y(T ) = 0. To solve this problem, we show the problem of null
controllability is equivalent to solving a moment problem of the form:
yn0 eλn T
=
bn
Z
T
w(τ )eλn τ dτ.
0
To prove the existence of the solution w(t) to the moment problem, we study the separability of
the eigenvalues λn of the system and obtain estimates of the Fourier coefficients bn of the control
operator B. We can then apply results on biorthogonal sequences developed by Russell and Fattorini
in [?]. It is important to show eigenvalue separation. These eigenvalues depend in a complex way
on the parameters of the system, and separate arguments are needed for the cases where they are
rationally and irrationally related. In either case we obtain the necessary bounds to show separation
of the eigenvalues and the existence of a biorthogonal sequence when applying a Dirichlet control. In
addition, we have also attempted the case of applying Neumann controls in the particular case that
the interval is symmetric and all coefficients are equal to one. We also obtain null controllability
with the difference that eigenfunctions are smaller to one side of the point mass. I also have begun
studying the case where two point-masses are introduced in the interior. Alongside these results,
it is important to justify the physics of the system. The modeling of such systems can be thought
of as the limit of a sequence of PDE depending on a parameter > 0. More precisely, we consider
a system modeling heat flow of a thin rod, occupying the interval (−1 − , 1 + ) with the middle
section (−, ) having a specific heat equal to 1/2. Using techniques described by Lagnese and
Lions in [?] and by Castro in [?], I intend to show weak convergence of the solutions.
In addition to my current research problem, I have worked on solutions to soliton equations while
I attended my undergraduate institution and for a year at Kobe University in Japan. Furthermore,
on the summer of 2014 I attended the IMA Math Modeling Workshop at the University of British
Columbia where we worked for a week on Fast Calculation of Diffraction by Photomasks under the
guidance of Dr. Apo Sezginer of KLA-Tencor.
My future work will consist of completing the null controllability of the heat equation with an
interior point mass for arbitrary parameters of specific heat, density, conductivity and length of the
rod. Once null controllability is established, I expect approximate controllability to be equivalent.
The next step would be to complete the physical justification of the model. This step will consist
of proving weak convergence of the solutions. Later, the possibility of strong convergence may be
explored.
Once the thermal system is understood, I intend to combine the known results of the wave
equation and that of the heat equation to understand the thermoelastic system with an interior
point mass. I expect the physical justification to be a superposition of the heat and wave equation
models. The controllability of the system is more challenging and interesting. Furthermore, since
the spectrum of the heat equation is related to that of Schödinger’s equation, I would like to expand
existing results.
References
[1] S. Hansen and E. Zuazua, Exact controllability and stabilization of a vibrating string with an interior point mass,
SIAM J. Cont. Optim. 33(5), 1995 1357–1391.
[2] A. O. Ozer, Exact boundary controllability and feedback stabilization for a multi-layer Rao-Nakra beam, Ph.D.
Thesis, Iowa State University.
[3] C. Castro, Asymptotic analysis and control of a hybrid system composed by two vibrating strings connected by a
point mass, ESAIM Control Optim. Calc. Var. 2 (1997) 231–280.
[4] C. Castro and E. Zuazua, A hybrid system consisting in two flexible beams connected by a point mass: Spectral analysis and well-posedness in asymmetric spaces, J. Blum, A. Raoult (Eds.), Elasticité, Viscoélasticité et
Contrôle Optimal, ESAIM Proc. vol. 2, 1997, pp. 17–53.
[5] J. E. Lagnese and J. -L. Lions, Modeling, Analysis and Control of Thin Plates, Collection RMA, vol. 6 Masson,
Paris (1988)
[6] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts,
Birkhäuser, Basel, Switzerland, 2009.
[7] R. M. Young, An introduction to non-harmonic Fourier series, Bulletin (New Series) of the American Mathematical Society 5 (1981), no. 3, 324–329.
[8] D. Russell and H. O. Fattorini, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rat. Mech. Anal. 43 (1971), 272–292.
[9] D. Russell and H. O. Fattorini, Uniform bounds on biorthogonal functions for real exponentials with an application
to the control theory of parabolic equations, Quart. Appl. Math. 43 (1974), 45-69.
[10] D. Russell, Non-harmonic Fourier series in the control theory of distributed parameter systems J. Math. Anal.
Appl. 18 (1967), 542–560.