Title of the course: Well-Posedness and Control of Evolution PDEs with Inhomogeneous Boundary Conditions
Instructor: Assoc. Prof. Dionyssios Mantzavinos, Türker Özsarı
Institution: University of Kansas, Bilkent Ü.
Dates: 31 August – 6 September 2026
Prerequisites: A familiarity with basic-level complex variables and introductory material on partial differential equations would be useful.
Level: Advanced undergraduate and graduate
Abstract: In this course we will study well-posedness and boundary control for a class of evolution partial differential equations (PDEs) with inhomogeneous boundary conditions, motivated by models from wave propagation and dispersive dynamics. Core examples will include the wave equation and dispersive equations such as the nonlinear Schrödinger and Korteweg–de Vries equations; as time allows, we may also discuss related higher-order models.
We will present two approaches to the well-posedness of weak solutions for initial-boundary value problems with inhomogeneous boundary conditions. The first approach is the method of transposition, in which solutions are defined in a dual (distributional) sense through testing against smooth solutions of the adjoint problem. Rather than enforcing boundary conditions pointwise, boundary forcing is incorporated via the adjoint traces that arise from integration by parts. This perspective is particularly well suited to handling rough boundary and initial data and to obtaining existence, uniqueness, and stability estimates in natural energy spaces on general domains. Within the same framework, we will discuss basic controllability theory via HUM (Hilbert Uniqueness Method) in a strictly duality-based form: controllability is characterized through an observability inequality for the adjoint system, and the control is constructed explicitly from adjoint solutions (with the boundary trace playing the role of the control input). In particular, we will emphasize how the transposition framework provides the right setting to interpret boundary controls and to justify the control-to-state map when the control has low regularity.
The second approach uses a combination of fundamental tools and techniques from complex analysis, collectively known as the unified transform (or Fokas method). Assuming the existence of sufficiently regular solutions, we will derive analytic solution representations for various initial–boundary value problems associated with the aforementioned linear evolution PDEs. The resulting formulae involve integrals of the initial and boundary data along suitable contours in the complex plane. Importantly, we will show that these integrals remain meaningful under very weak assumptions on the data, thereby yielding weak solutions in a much broader (low-regularity) setting. We will then use these unified-transform representations to establish well-posedness of the corresponding nonlinear PDEs in appropriate function spaces, in the sense of Hadamard (existence, uniqueness, and continuous dependence on the data). In particular, we will present a novel approach for proving linear spatiotemporal estimates, which are central to the analysis of the associated nonlinear problems. Finally, we will show how these representation formulae can also be used to prove certain control theoretical results.
Language: EN