# Analysis and Partial Differential Equations Unit (Ugur Abdulla)

The aim of the Analysis and Partial Differential Equations (PDE) unit is to reveal and analyze the mathematical principles reflecting natural phenomena expressed by partial differential equations. Research focuses on fundamental analysis of PDEs, regularity theory of elliptic and parabolic PDEs, with special emphasis on the regularity of finite boundary points and the point at \(\infty\), its measure-theoretical, probabilistic and topological characterization, well-posedness of PDE problems in domains with non-smooth and non-compact boundaries, global uniqueness, analysis and classification of singularities, asymptotic laws for diffusion processes, regularity theory of nonlinear degenerate and singular elliptic and parabolic PDEs, free boundary problems, optimal control of free boundary systems with distributed parameters. Current areas of interest include Potential Theory, Harmonic Analysis, Probability Theory, Calculus of Variations and Optimal Control, Optimization, Mathematical Biosciences and Quantum Biology. Some of the current research projects in Applied Mathematics include laser ablation of biomedical tissues; preventing aerodynamic stall by in-flight ice accretion in the aerospace industry; cancer detection through Electrical Impedance Tomography and optimal control theory; identification of parameters in large-scale models of systems biology; optimal control of reactive oxygen species in quantum biology.

**1. Potential Theory and PDEs**

**1.1. Kolmogorov Problem and Wiener Criterion at ****∞ **

In 1928, Kolmogorov posed the following open problem in the seminar on probability theory at the Moscow State University. Given function \(h\in C(-\infty,0]\) such that \(h(0)=0, h(-\infty)=+\infty, \ h\in \uparrow, \ (-t)^{\frac{1}{2}} h\in \uparrow,\) as \( t\downarrow -\infty,\) let \(u_n(x,t), n=1,2,...\) be a sequence of solutions of the initial boundary value problems for the one-dimensional heat equation in expanding domains: \(u_t = u_{xx} \quad\text{in} \ |x|< h(t), \ u=0 \quad\text{on} \ |x|= h(t), -n < t < 0; \ u|_{t=-n}=1\)

From the maximum principle, it follows that \(u_n(x,t) \ge u_{n+1}(x,t) \ge u(x,t)=\lim\limits_{n\to +\infty}u_n(x,t) \ge 0, \ (x,t)\in \mathbb{R}\times (-\infty,0).\)

** Kolmogorov Problem**: Is \(u(x,t)=\) or \(>0?\)

Kolmogorov’s motivation for posing this problem was a connection to the probabilistic problem of finding asymptotic behavior at infinity of the standard Brownian path. Let \(\{\xi(t): t\geq 0, P_{\bullet}\}\) be a standard 1-dimensional Brownian motion and \(P_{\bullet}({\bf B})\) is the probability of the event \({\bf B}\) as a function of the starting point \(\xi(0)\). Blumenthal’s 01 law implies that \(P_0 \{\xi(t)< h(-t), t\uparrow +\infty\}=0\) or 1; \(h(-t)\) is said to belong to upper class if this probability is 1 and to the lower class otherwise. Remarkably, ** Kolmogorov Problem**’s solution \(u\) is \(=0\) or \(>0\) according to as \(h(-t)\) is in lower or upper class accordingly.

**in a one-dimensional setting was solved by Petrovski in 1935, and the celebrated result is known as the**

*Kolmogorov Problem***in the probabilistic literature.**

*Kolmogorov-Petrovski test*** Kolmogorov Problem **is a particular case of the following fundamental problem in Analysis, PDEs, and Potential Theory. Let \(\Omega\subset\mathbb{R}^{N+1}\) be an arbitrary open set. Given bounded Borel measurable function \(f:\partial\Omega\to \mathbb{R}\), consider the parabolic Dirichlet problem

(1) \(u_t=\Delta u, \quad\text{in} \ \Omega; \ u=f \quad\text{on} \ \partial\Omega\)

** Problem \(\mathcal{AK}\)**:

*Is bounded solution of (1) unique without prescribing \(f\)*

*and \(u\)*

*at \(\infty\)*

*?*

In fact, the parabolic Dirichlet problem (1) has one and only one, or infinitely many bounded solutions without prescribing \(f\) and \(u\) at \(\infty\) [1]. ** Kolmogorov Problem** is a particular case of the

**Problem \(\mathcal{AK}\)**with \(\Omega=\Omega_h\) and \(f=0\).

** Kolmogorov Problem** (or

**Problem**\(\mathcal{AK}\)) is the fundamental problem for general second-order parabolic and elliptic PDEs. Next, is the formulation of the problem for the second-order elliptic PDEs. Given an open set \(\Omega\subset \mathbb{R}^N, N\geq 3\), and bounded Borel measurable function \(f:\partial\Omega\to \mathbb{R}\), consider the Dirichlet problem

(2) \({\cal A}u:=-(a_{ij}(x)u_{x_i})_{x_j}=0, \quad\text{in} \ \Omega; \ u=f \quad\text{on} \ \partial\Omega\)

where \({\cal A}\) is the uniformly elliptic operator with bounded measurable coefficients. Without prescribing \(f\) and \(u\) at \(\infty\) the Dirichlet problem (2) has one and only one or infinitely many solutions [3,4]. **Problem \(\mathcal{AK}\)** (or** Kolmogorov Problem**) is to find a necessary and sufficient condition on \(\Omega\) for the uniqueness of the bounded solution to the Dirichlet problem (2).

The full solution of the ** Kolmogorov Problem** (or

**Problem \(\mathcal{AK}\)**) is presented in [1,2] for the heat equation, and in [3,4] for the linear elliptic PDEs. A new concept of

*the regularity*or

*irregularity of*\(\infty\) is introduced according to as the

*caloric*(or \({\cal A}\)-

*harmonic*)

*measure*of \(\infty\) is null or positive, and the necessary and sufficient condition for the

**Problem \(\mathcal{AK}\)**is proved as Wiener-type criterion for the regularity of \(\infty\).

Norbert Wiener's celebrated result on the boundary regularity of harmonic functions is one of the most beautiful and delicate results of 20th-century mathematics. It has shaped the boundary regularity theory for elliptic and parabolic PDEs and has become a central result in the development of potential theory at the intersection of functional analysis, PDE, and measure theories. Full solution of the ** Kolmogorov Problem** and new Wiener criterion for the regularity of the point at \(\infty\) for second-order elliptic and parabolic PDEs broadly extend the role of the Wiener test in classical analysis. The Wiener test at \(\infty\) arises as a global characterization of uniqueness in boundary value problems for arbitrary unbounded open sets. From a topological point of view, the Wiener test at \(\infty\) arises as a thinness criterion at \(\infty\) in fine topology. In a probabilistic context, the Wiener test at \(\infty\) characterizes asymptotic laws for the characteristic Markov processes whose generator is the given differential operator. The counterpart of the new Wiener test at a finite boundary point leads to uniqueness in the Dirichlet problem for a class of unbounded functions growing at a certain rate near the boundary point; a criterion for the removability of singularities and/or for unique continuation at the finite boundary point.

- U. G. Abdulla, Wiener’s Criterion at ∞ for the Heat Equation, Advances in Differential Equations, 13, 5-6(2008), 457-488.
- U. G. Abdulla, Wiener's Criterion at ∞ for the Heat equation and its Measure-theoretical Counterpart, Electronic Research Announcements in Mathematical Sciences, 15 (2008), 44-51.
- U. G. Abdulla, Wiener’s Criterion for the Unique Solvability of the Dirichlet Problem in Arbitrary Open Sets with Non-Compact Boundaries, Nonlinear Analysis, 67, 2(2007), 563-578.
- U. G. Abdulla, Regularity of ∞ for Elliptic Equations with Measurable Coefficients and Its Consequences, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 32, 10(2012), 3379-3397.

**1.2. Analysis of Singularities for the Elliptic and Parabolic PDEs and Asymptotic Laws for Markov Processes**

One of the main problems in the Analysis of PDEs is understanding the nature of singularities of solutions of PDEs reflecting the natural phenomena. Solution of the Kolmogorov Problem and new Wiener-type criterion for the regularity of \(\infty\) opens a great new perspective for the breakthrough in understanding non-isolated singularities performed by solutions of second-order elliptic and parabolic PDEs at the finite boundary points and the point at \(\infty\).

In a recent paper [5], the idea is implemented to prove the Wiener criterion for the removability of the logarithmic singularity for the second-order elliptic PDEs in \(\mathbb{R}^2\). Let \(D=\{x\in \mathbb{R}^2: 0<|x|<1 \}\), and \(h\) is a Green function of the elliptic operator \({\cal A}\) from (2) on the unit disk with the pole at the origin \(\zeta\) ( \(h=-log|x|\) if \({\cal A}=-\Delta\)). The Dirichlet problem (2) on \(D\) has infinitely many solutions in a class \(O(h)\) at \(\zeta\), or otherwise speaking logarithmic singularity at \(\zeta\) is not removable. Let \(\Omega\subset D\) be an open subset and \(f=O(h)\) at \(\zeta\) . The major open problem is to find a criterion on \(\Omega\subset D\) for the unique solvability of the Dirichlet problem (2) in a class \(O(h)\) at \(\zeta\), without further specification of the behavior of solution at \(\zeta\). Equivalently, the aim is to find a necessary and sufficient condition on \(\Omega\) for the removability of the logarithmic singularity at \(\zeta\) for the elliptic PDE. An equivalent problem can be posed in an open subset of \(D^c\) near the point at \(\infty\) through inversion of \(\zeta\).

The major problem can be reformulated in the frame of \(h\)-harmonic functions. A function \(v/h\) is called \({\cal A}_h\)-harmonic if \(v\) is \({\cal A}\)-harmonic, i.e. \({\cal A}u=0\) in \(\Omega\). Given a bounded Borel measurable function \(g\) on \(\partial\Omega\), consider \(h\)-Dirichlet problem (\(h\)-DP) to find a \({\cal A}_h\)-harmonic function such that \(u=g\) on \(\partial\Omega\setminus\{\zeta\}\). The following is the analogous Kolmogorov Problem.

** Problem \(\mathcal{AK}_h\)**:

*Is a bounded solution*

*\(u\) of the \(h\)*-DP

*unique without prescribing*\(g\) and \(u\) at \(\zeta\)?

\(h\)-DP has one and only one, or infinitely many bounded solutions [5]. In particular, if \(g=f=0\), ** Problem \(\mathcal{AK}_h\) **asks if 0 is the only solution of (2) in class \(O(h)\), or equivalently, whether or not logarithmic singularity is removable. It is a fundamental problem bridging several fields of Analysis, PDEs, and Potential Theory. Define \({\cal A}_h\)-harmonic measure of \(\{\zeta\}\) as \(\mu^h_{\Omega}(\cdot, \zeta)= \ ^hH_{1_\zeta}^{\Omega}(\cdot),\) the latter is a Perron solution of the \(h\)-DP with boundary function being an indicator function \(1_\zeta\). Measure-theoretical counterpart of the

**is whether \(\cal{A}_h\)-harmonic measure of \(\{\zeta\}\) is null or positive. Topological counterpart of the**

*Problem \(\mathcal{AK}_h\)***is whether \(\Omega\) is a deleted neighborhood of \(\zeta\) in \(\cal{A}_h\)-minimal fine topology, or equivalently \(\Omega^c\cap D\) is \(\cal{A}_h\)-minimal thin at \(\zeta\).**

*Problem \(\mathcal{AK}_h\)*In the probabilistic context, ** Problem \(\mathcal{AK}_h\)** expresses asymptotic properties of the conditional Markov processes associated with the corresponding differential operator: let \({\cal A}=-\Delta\), and consider \(h\)-Brownian motion on \(D\), where \(h(\xi)=-log|\xi|\). It is known that almost every \(h\)-Brownian path \(w_\xi^h(t)\) from \(\xi\in D\) has a finite lifetime \(S_\xi^h\) and tends to \(\zeta\) at its lifetime. Let \({\bf B}\) be an event that \(\{t: w_\xi^h(t)\in \Omega^c\}\) clusters to \(S_\xi^h\). According to Kolmogorov’s \(01\)-law \(P({\bf B})=0\) or 1. One can also consider \(h\)-Brownian motion to start from \(\zeta\) - infinity of \(h\). In fact, it can be established with the standard procedure of Kolmogorov that \(h\)- Brownian trajectory \(w_\xi^h(t)\) in \(D\) from \(\zeta\) is a process whose paths have finite lifetimes and tend back to \(\zeta\) at this lifetimes. Let \(m_{\Omega^c}=\inf\{t: t>0, w_\zeta^h(t) \in \Omega^c\}\) be the Markovian exit time from \(\Omega\). Then \(P(m_{\Omega^c}=0)=0\) or \(1\). Similar questions can be formulated for the conditional Markov process with infinitesimal Dynkin generator being a differential operator \(-{\cal A}\).

** The probabilistic counterpart of the Problem \(\mathcal{AK}_h\)**: Given open set \(\Omega\subset D\), is \(P({\bf B})=0\) or 1? Is \(P(m_{\Omega^c}=0)=0\) or 1?

To solve ** Problem \(\mathcal{AK}_h\)**, paper [5] introduced a new concept of \(h\)-regularity (or \(h\)-irregularity) of \(\zeta\) for \(\Omega\), according to as whether \({\cal A}_h\)-harmonic measure of \(\zeta\) is null or positive. Paper [5] revealed a new concept of \(h\)-capacity and proved a necessary and sufficient condition for the

*Problem \(\mathcal{AK}_h\)***in terms of the Wiener criterion for the \(h\)-regularity (or \(h\)-irregularity) of \(\zeta\) for \(\Omega.\)**

One of the oldest problems in the theory of PDEs is the problem of finding geometric conditions on the boundary manifold for the regularity of the solution of the elliptic and parabolic PDEs. There is a connection between this problem and the problem of asymptotics of the corresponding Markov processes. In [6], a geometric iterated logarithm test for the boundary regularity of the solution to the heat equation is established. In addition, an exterior hyperbolic paraboloid condition for the boundary regularity, which is the parabolic analogy of the exterior cone condition for the Laplace equation is proved. In fact, for the characteristic top boundary point of the symmetric rotational boundary surfaces, the necessary and sufficient condition for the regularity coincides with the well-known Kolmogorov-Petrovski test for the local asymptotics of the multi-dimensional Brownian motion trajectories [7]. In the case when symmetric rotational boundary surfaces extend to \(\infty\), the regularity of the point at \(\infty\) precisely characterize the uniqueness of the bounded solutions. A geometric necessary and sufficient condition for the regularity of \(\infty\) is proved in [8]. In the probabilistic context, the result coincides with the Kolmogorov-Petrovski test for the asymptotics of the multi-dimensional Brownian motion trajectories at \(\infty\) [9].

- U. G. Abdulla, Removability of the Logarithmic Singularity for the Elliptic PDEs with Measurable Coefficients and its Consequences, Calculus of Variations and Partial Differential Equations Volume 57:157, December 2018.
- U. G. Abdulla, First Boundary Value Problem for the Diffusion Equation. I. Iterated Logarithm Test for the Boundary Regularity and Solvability, SIAM J. Math. Anal., 34, 6, 2003, 1422-1434.
- U. G. Abdulla, Multidimensional Kolmogorov-Petrovsky test for the boundary regularity and irregularity of solutions to the heat equation, Boundary Value Problems, 2, 2005, 181-199.
- U. G. Abdulla, Necessary and sufficient condition for the existence of a unique solution to the first boundary value problem for the diffusion equation in unbounded domains, Nonlinear Analysis, 64, 5(2006), 1012-1017.
- U. G. Abdulla, Kolmogorov problem for the heat equation and its probabilistic counterpart, Nonlinear Analysis, 63, 5-7, 2005, 712-724.

**2. Qualitative Theory and Regularity for Nonlinear PDEs**

PDEs arising in a majority of real-world applications are nonlinear. Despite its complexity, in the theory of nonlinear PDEs, one can observe key equations or systems which are essential for the development of the theory for a particular class of equations. One of those key equations is the nonlinear degenerate/singular parabolic equation, so-called nonlinear diffusion equation (NDE):

\(u_t=\Delta u^m\)

where \(u=u(x,t), x\in\mathbb{R}^N, t\in\mathbb{R}, m>0, m\neq 1\). The NDE is nonlinear degenerate (\(m>1\)) or singular (\(0 < m<1\)) parabolic equation arising in various applications such as flow of a gas or Newtonian fluid through porous media, heat conduction in a plasma, growth of the cancerous tumor, etc. Despite the formal similarity with the classical heat equation (\(m=1\)), both qualitative and regularity theory of the NDE is significantly different. In particular, NDE with \(m>1\) possesses the finite speed of propagation property (FSPP) contrasting the infinite speed of propagation property of the linear diffusion equation. FSPP confirms the physical relevance of NDE. Solutions with compactly supported initial perturbations remain so for all time, and they fail to be differentiable on the *interfaces or free boundaries* which separate the positivity set from the "empty region" where the solution is identically zero. The latter happens due to implicit degeneration of the equation, or cease of parabolicity of the PDE. Therefore, qualitative and regularity theory for NDE-type PDEs is quite different from classical parabolic PDEs.

The fundamental question on the boundary regularity and general theory of nonlinear degenerate and singular parabolic PDEs in general non-smooth and non-cylindrical domains under the minimal regularity assumptions on the boundary manifolds was solved in [10-12]. Remarkably, the boundary regularity of the solution is expressed in terms of the one side Hölder condition on the boundary manifold, or equivalently in terms of the Hölder regularity of the exterior touching surface, and the critical value of the Hölder exponent is, independently of \(m\), equal to \(1/2\). This development was motivated by numerous applications. Primarily by applying the corresponding one-dimensional theory [13,14], one of the old problems (posed by Barenblatt in 1952) on the classification of the short-time behavior of interfaces for the reaction-diffusion equations was solved in [15,16]. The methods of [15,16] were successfully applied to solve the interface problem for various classes of reaction-diffusion type equations in [18-21].

- U. G. Abdulla, Well-posedness of the Dirichlet Problem for the Nonlinear Diffusion Equation in Non-smooth Domains, Transactions of Amer. Math. Soc., 357, 1, 2005, 247-265.
- U.G. Abdulla, Reaction-diffusion in nonsmooth and closed domains, Boundary Value Problems, Special issue: Harnack estimates, Positivity and Local Behavior of Degenerate and Singular Parabolic Equations, Vol. 2007 (2007).
- U.G. Abdulla, On the Dirichlet problem for the nonlinear diffusion equation in non-smooth domains, J. Math. Anal. Appl., 246, 2, 2001, 384-403.
- U.G. Abdulla, Reaction-diffusion in Irregular Domains, J. Differential Equations, 164, 2000, 321-354.
- G. Abdulla, Reaction-diffusion in a closed domain formed by irregular curves, J. Math. Anal. Appl., 246, 2000, 480-492.
- U.G. Abdulla, J.R. King, Interface Development and Local Solutions to Reaction-Diffusion Equations, SIAM J. Math. Anal., 32, 2, 2000, 235-260.
- U.G. Abdulla, Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption, Nonlinear Analysis, 50, 4, 2002, 541-560.
- U.G. Abdulla, Local structure of solutions of the Dirichlet problem of N-dimensional reaction-diffusion equations in bounded domains, Advance in Differential Equations, Volume 4, Number 2, 1999, 197-224.
- U.G. Abdulla, R. Jeli, Evolution of interfaces for the nonlinear parabolic p-Laplacian type reaction-diffusion equations, European Journal of Applied Mathematics, Volume 28, Issue 5, 2017, 827-853.
- U.G. Abdulla, J. Du, A. Prinkey, Ch. Ondracek, S. Parimoo, Evolution of Interfaces for the Nonlinear Double Degenerate Parabolic Equation of Turbulent Filtration with Absorption, Mathematics and Computers in Simulation, 153 (2018), 59-82.
- U.G. Abdulla, R. Jeli, Evolution of interfaces for the non-linear parabolic p -Laplacian type reaction-diffusion equations. II. Fast diffusion vs. strong absorption. European Journal of Applied Mathematics, Volume 31, Issue 3, 2019, 385-406.
- U.G. Abdulla, A. Abuweden, Interface development for the nonlinear degenerate multidimensional reaction-diffusion equations, Nonlinear Differential Equations and Applications NoDEA, February 2020, 27:3.
- U.G. Abdullaev, Instantaneous shrinking of the support of a solution of a nonlinear degenerate parabolic equation. (Russian) Mat. Zametki 63(1998), no.3,323-331; translation in Math. Notes 63(1998), no.3-4,285-292.
- U. G. Abdullaev, On sharp local estimates for the support of solutions in problems for nonlinear parabolic equations. (Russian) Mat. Sbornik 186(1995), no.8,3-24; translation in Sb. Math. 186(1995), no.8,1085-1106.

**3. Optimal Control of Free Boundary Systems with Distributed Parameters**

The project aims to advance the boundary of knowledge in optimal control of free boundary systems with distributed parameters described by second-order linear and nonlinear parabolic PDEs. It addresses a broad class of optimal control and inverse free boundary problems with distributed parameters describing phase transition processes of Stefan type arising in thermo-physics, fluid mechanics, biomedical engineering, and various nonlinear diffusion problems such as chemical reaction-diffusion, diffusion in a porous media, diffusion of biological populations. Specifically, the project is motivated by two applied problems:

- Optimal control and inverse problems for the identification of parameters in biomedical problem on laser ablation of biological tissues;
- Preventing aerodynamic stall in aircraft by in-flight ice accretion via optimal control of supercooled Stefan free boundary problem;

In [24,25] a new variational formulation of the inverse one-phase Stefan problem is developed, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. An optimal control framework is employed, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consist of the minimization of the sum of \(L_2\)-norm declinations from the available measurement of the temperature flux on the fixed boundary and available information on the phase transition temperature on the free boundary. This approach allows one to tackle situations when the phase transition temperature is not known explicitly and is available through measurement with possible error. In [24,25] the existence of the optimal control and the convergence of the discretized optimal control problems both with respect to cost functional and control is proved. Frechet differentiability in Besov spaces framework and optimality condition in the new optimal control framework was proved in [26,27], an iterative gradient method for the numerical solution was implemented in [28,29].

The new variational approach developed in [24,25] does not apply to the multiphase Stefan problem. In [30,31] a new approach was developed based on the weak formulation of the multiphase Stefan problem as a boundary value problem for the nonlinear PDE with a discontinuous coefficient. In [30] the optimal control framework was applied to the inverse multiphase Stefan problem with non-homogeneous Neumann conditions on the fixed boundaries in the case when the space dimension is one. The control vector was taken to be the heat flux on the left boundary and the optimality criteria consisted of the \(L_2\)-norm declinations from a measurement of the temperature on the right fixed boundary. The full discretization was implemented and convergence of the discrete optimal control problems to the original problem was proved. In [31] the new method was developed for the inverse multidimensional and multiphase Stefan problem with homogeneous Dirichlet boundary condition on a bounded Lipschitz domain, where the density of the heat source is unknown in addition to the temperature and the phase transition boundaries. In [32], the results of [30] are extended to the case of general second-order singular parabolic PDEs.

- U. G. Abdulla On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations. I. Well-posedness and Convergence of the Method of Lines, Inverse Problems and Imaging, Volume 7, Number 2(2013), 307-340.
- U. G. Abdulla On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations: II. Convergence of the Method of Finite Differences, Inverse Problems and Imaging, Volume 10, Number 4(2016), 869-898.
- U.G. Abdulla and J. Goldfarb Frechet Differentiability in Besov Spaces in the Optimal Control of Parabolic Free Boundary Problems, Inverse and Ill-posed Problems, Volume 26, Issue 2, 2018, 211-228.
- U. G. Abdulla, E. Cosgrove, J. Goldfarb On the Frechet Differentiability in Optimal Control of Coefficients in Parabolic Free Boundary Problems, Evolution Equations and Control Theory, Volume 6, Issue 3, 2017, 319-344.
- U. G. Abdulla, V. Bukshtynov, A. Hagverdiyev Gradient Method in Hilbert-Besov Spaces for the Optimal Control of Parabolic Free Boundary Problems, Journal of Computational and Applied Mathematics, Volume 346, January 2019, 84-109.
- U. G. Abdulla, J. Goldfarb and A. Hagverdiyev Optimal Control of Coefficients in Parabolic Free Boundary Problems Modeling Laser Ablation, Journal of Computational and Applied Mathematics, Volume 372, July 2020.
- U. G. Abdulla and B. Poggi Optimal control of the multiphase Stefan problem, Applied Mathematics & Optimization, 80, 2(2019), 479-513.
- U. G. Abdulla and B. Poggi Optimal Stefan Problem, Calculus of Variations and Partial Differential Equations, 59, 61(2020).
- U. G. Abdulla and E. Cosgrove, Optimal Control of Multiphase Free Boundary Problems for Nonlinear Parabolic Equations, Applied Mathematics and Optimization, 84, 2021, 589-619.

**4. Mathematical Biosciences and Quantum Biology**

**4.1. Identification of Parameters in Systems Biology**

Systems Biology is an actively emerging interdisciplinary area between biology and applied mathematics, based on the idea of treating biological systems as a whole entity which is more than the sum of its interrelated components. These systems are networks with emerging properties generated by a complex interaction of a large number of cells and organisms. One of the major goals of systems biology is to reveal, understand, and predict such properties through the development of mathematical models based on experimental data. In many cases, predictive models of systems biology are described by large systems of nonlinear differential equations. Quantitative identification of such systems requires the solution of inverse problems on the identification of parameters of the system. The main focus of the recent paper [33] is the inverse problem for the identification of parameters for systems of nonlinear ODEs arising in systems biology. A new numerical method that combines Pontryagin optimization, Bellman's quasi-linearization with sensitivity analysis, and Tikhonov's regularization is implemented. The method is applied to various biological models such as the Lotka-Volterra system, bistable switch model in genetic regulatory networks, gene regulation, and repressilator models from synthetic biology. The numerical results and application to real data demonstrate that the method is very well adapted to canonical models of systems biology with moderate size parameter sets and has quadratic convergence. Software package *qlopt* is developed to implement the method and posted in GitHub under the GNU General Public License v3.0.

To address adaptation and the scalability of the method to inverse problems with significantly larger size of parameter sets a modification of the method is developed in [34] by embedding a method of staggered corrector for sensitivity analysis and by enhancing multi-objective optimization which enables the application of the method to large-scale models with practically non-identifiable parameters based on multiple data sets, possibly with partial and noisy measurements. Application of the modified method to benchmark models demonstrates the super-linear convergence and the advantage of *qlopt* over most popular methods/software such as *lsqnonlin, fmincon *and* nl2sol*.

- U. G. Abdulla and R. Poteau, Identification of Parameters in Systems Biology, Mathematical Biosciences, Volume 305, November 2018, 133-145
- U. G. Abdulla and R. Poteau, Identification of Parameters for Large-scale Kinetic Models, Journal of Computational Physics, Volume 429, 2021, Paper No. 110026, 19 pp.

**4.2. Cancer Detection through Electrical Impedance Tomography (EIT) and Optimal Control Theory**

The goal of this project is to develop a new mathematical framework utilizing the theory of PDEs, inverse problems, and optimal control of systems with distributed parameters for the better understanding of the development of cancer in the human body, as well as the development of effective methods for the detection and control of tumor growth.

In a recent paper [35], the inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the electrodes for a given electrode current is analyzed. A PDE constrained optimal control framework in Besov space is developed, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm difference of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The novelty of the control-theoretic model is its adaptation to the clinical situation when additional "*voltage-to-current*" measurements can increase the size of the input data from \(m\) up to \(m!\) while keeping the size of the unknown parameters fixed. The existence of the optimal control and Fréchet differentiability in the Besov space along with optimality condition is proved. Numerical analysis of the simulated model example in the 2D case demonstrates that by increasing the number of input boundary electrode currents from \(m\) to \(m^2\) through additional "*voltage-to-current*" measurements the resolution of the electrical conductivity of the body identified via gradient method in Besov space framework is significantly improved. In [36], the EIT optimal control problem is fully discretized using the method of finite differences. New Sobolev-Hilbert space is introduced, and the convergence of the sequence of finite-dimensional optimal control problems to EIT coefficient optimal control problem is proved both with respect to functional and control in 2- and 3-dimensional domains.

- U. G. Abdulla, V. Bukshtynov, S. Seif, Cancer Detection through Electrical Impedance Tomography and Optimal Control Theory: Theoretical and Computational Analysis, Volume 18, Issue 4, 2021, 4834-4859.
- U. G. Abdulla, S.Seif, Discretization and Convergence of the EIT Optimal Control Problem in 2D and 3D Domains, to appear in Inverse and Ill-posed Problems, 2022.

**4.3 Optimal Control of Reactive Oxygen Species in Quantum Biology**

One of the greatest challenges in the field of chemical and physical biology is to bridge the knowledge gap between the atomic level and the cellular level. Focused on the biological quantum/classical interface, an emerging field called Quantum Biology has promised to offer new and compelling insights into fundamental underlying cellular processes from the perspective of quantum phenomena.

The overarching goal of this project is to develop and exploit the advanced methods of quantum optimal control theory to reveal the deep relationship between functional optimization of the internal hyperfine parameters in flavoproteins and/or external magnetic field intensity input to maximize singlet quantum yield in biochemical processes. This represents a completely new area of Quantum Biology and the proposed work will unveil a groundbreaking role of quantum coherence in biochemical processes.

In a forthcoming recent paper [37], the coherent spin dynamics of radical pairs in biochemical reactions modeled by the Schrödinger system with spin Hamiltonians given by the sum of Zeeman interaction and hyperfine coupling interaction terms is analyzed. We considered the problem of identification of the constant external magnetic field and internal hyperfine parameters which optimize the quantum singlet-triplet yield of the radical pair system. Based on the recent development on the identification of parameters in Systems Biology [33], and in large-scale kinetic models [34], we developed the *qlopt* algorithm to identify optimal values of 3-dimensional external electromagnetic field vector and 3- or 6-dimensional hyperfine parameter vector which optimize the quantum singlet-triplet yield for the spin dynamics of radical pairs in 8- or 16-dimensional Schrödinger system corresponding to one- and two-proton cases respectively. Numerical results demonstrate that the quantum singlet-triplet yield of the radical pair system can be significantly reduced if optimization is pursued simultaneously for external magnetic field and internal hyperfine parameters. The results help us understand the structure-function relationship of the putative magnetoreceptor to manipulate and enhance quantum coherences at room temperature, and leverage biofidelic function to inspire novel quantum devices. In particular, the results provide new routes for weak biomagnetic sensors. The results represent a crucial step to affirm the direct connection between hyperfine optimization and quantum coherence.

The objective of the recent paper [38] is to develop the mathematical foundation of quantum optimal control with the ultimate goal of understanding fundamental mechanisms connecting atomic and cellular levels in Quantum Biology. Precisely, the optimal control of the external electromagnetic field and internal hyperfine parameters for the minimization of the quantum singlet-triplet yield of the radical pairs in biochemical reactions modeled by Schrödinger system with spin Hamiltonians given by the sum of Zeeman interaction and hyperfine coupling interaction terms will be analyzed. We hypothesize that Pontryagin's Maximum Principle and optimal state vector with bang-bang optimal control presents the mathematical counterpart of the quantum coherence concept, i.e. the ability of the system to exist in several distinct states simultaneously, such as the inter-system crossing between singlet-triplet states. The ultimate goal is to prove Pontryagin's maximum principle and reveal the bang-bang structure of the optimal control, develop and implement numerical algorithms and software packages based on theoretically proved optimality conditions, and form a computational foundation for experimental applications.

- C. Martino, P. Jimenez, J. Goldfarb, U.G. Abdulla, Identification of parameters in coherent spin dynamics of radical pairs in quantum biology, PLOS One, in review.
- U. G. Abdulla and C. Martino, Bang-bang optimal control in coherent spin dynamics of radical pairs in quantum biology, in preparation.