When Engineers Design the Machine and Its Brain Together

A New Way to Design Systems That Change Over Time and Space

In 2018, engineers at a chemical plant faced a stubborn problem. Their reactor kept oscillating unpredictably, wasting energy and threatening safety. The controller they had designed was mathematically sound—but it assumed certain physical properties of the system that turned out to be wrong. By the time they realized the mismatch, millions had been spent redesigning the hardware.

The issue wasn't poor engineering. It was a fundamental limitation in how engineers traditionally approach system design: you build the system, then you design the controller. These two steps happen separately, in sequence, by different teams, using different tools. And that sequential approach inevitably leaves performance on the table.

A pair of researchers from the Indraprastha Institute of Information Technology in New Delhi, Antika Yadav and Prasad Vilas Chanekar, have now developed a mathematical framework that could change this paradigm—particularly for a important class of systems governed by what are called parabolic partial differential equations. These are systems where something—temperature, chemical concentration, stress—varies continuously across space and changes over time, like heat diffusing through a metal rod or pollutants spreading through groundwater. The new approach, published in July 2026, proposes a way to design both the physical system and its controller simultaneously, in a unified mathematical framework that guarantees stability while optimizing performance.

The key insight is deceptively simple: instead of designing a controller for a fixed system, why not let the system itself be a design choice? If you're building a thermal regulation system, perhaps the thickness of the insulation, the material properties, and the control gains can all be tuned together to achieve the best possible outcome. The mathematics for doing this simultaneously—called control co-design, or CCD—has existed for simpler systems. But for the infinite-dimensional, spatially distributed systems that govern much of the physical world, the problem has been intractable.

"This is a fundamental extension of control co-design theory to the PDE domain," the researchers write. "While CCD has been extensively studied for lumped-parameter systems, its extension to PDE systems is largely unexplored."

The Science

To understand what Yadav and Chanekar have accomplished, it helps to first appreciate why PDEs pose such a challenge—and why the standard engineering approach of "design first, control second" has persisted despite its known inefficiencies.

Consider a simple example: a metal rod being heated at one end. The temperature at any point along the rod depends on both position (where along the rod you're measuring) and time (when you measure it). The mathematical object that captures this is a function of two variables, which means it can't be described by an ordinary differential equation, or ODE. An ODE deals with a single variable changing—think of a ball's position as a function of time. But heat flowing through a rod requires tracking how temperature varies across space at each moment, then how that spatial distribution changes moment by moment. That's fundamentally harder.

Partial differential equations are the language of continuum mechanics, fluid dynamics, thermodynamics, and much of materials science. They govern the behavior of essentially anything that varies continuously through space and evolves over time. The parabolic class of PDEs—which includes the heat equation and reaction-diffusion systems—is particularly common in engineering applications.

The traditional engineering response to this complexity has been to sidestep it. Rather than trying to control the PDE directly, engineers discretize space: they break the rod into a finite number of points, treat each point as a separate variable, and reduce the problem to a system of ODEs. This discretization was once the only practical approach for computation, and it still works reasonably well. But it introduces an approximation that can accumulate into significant errors—and more fundamentally, it obscures the infinite-dimensional nature of the underlying physics.

In their paper, Yadav and Chanekar work directly with the parabolic PDE and derive what they call "sufficient stability conditions." This is a set of mathematical constraints—inequalities involving the system's physical parameters and controller gains—that, if satisfied, guarantee the system will behave well. It won't oscillate wildly, diverge to infinity, or fail to reach equilibrium. The conditions are "sufficient" rather than "necessary" because they are conservative: they might rule out some designs that would actually work, but they never admit a design that definitely won't.

The stability analysis uses a technique from Lyapunov theory, a branch of control theory named after Russian mathematician Alexander Lyapunov. The core idea is to find a function of the system state—called a Lyapunov function—whose value decreases over time whenever the system is not at equilibrium. If such a function exists, the system must be stable. Yadav and Chanekar construct an explicit Lyapunov functional for the boundary-controlled parabolic PDE and show that it decays under certain conditions.

The derivation is technical, but the result is concrete. The sufficient conditions involve three inequalities that couple the system parameter (a diffusion coefficient related to how quickly the quantity spreads) with the boundary feedback gains (the control parameters that determine how the system reacts to what it measures at its edges).

Once the stability conditions are established, the researchers tackle the optimization problem itself. The goal is to minimize a cost function that balances two competing objectives: keeping the system close to its desired state (like maintaining a target temperature) and keeping the control effort small (using less energy). This is the standard quadratic cost functional from optimal control theory.

But there's a twist: in addition to optimizing over the control gains, the designers can also choose the physical parameters of the system. Should the thermal rod be made of copper or aluminum? Should it be longer or shorter? The paper frames these as design variables that can be optimized alongside the controller.

The problem, however, remains computationally formidable. The PDE constraint makes the optimization infinite-dimensional, and the cost functional involves integrals over both space and time. To make this tractable, Yadav and Chanekar discretize the spatial domain using finite differences—a standard numerical method for approximating PDEs. This converts the continuous problem into a system of ordinary differential equations with matrix coefficients.

The crucial step is converting the infinite-horizon integral cost into a trace expression involving the solution of a matrix equation (specifically, a Lyapunov equation). This algebraic reformulation transforms the problem from one requiring numerical integration over infinite time into one requiring only the solution of a time-independent matrix equation at each iteration of the optimization.

The researchers then develop a gradient-based algorithm to solve the resulting constrained optimization problem. They compute explicit expressions for the gradients of the cost with respect to both the design variables and the control gains—gradients that involve solving additional Lyapunov-like equations. This allows the use of efficient numerical optimization methods that iteratively improve the design.

What They Found

The paper's primary contribution is theoretical: a formulation and solution procedure for the control co-design problem for a class of parabolic PDE systems with Neumann boundary feedback. But the researchers validate their approach through a detailed numerical example that demonstrates the efficacy of the proposed method across different types of co-design problems.

The numerical studies examine four distinct scenarios:

Case 1: Simultaneous optimization of both physical parameters. Here the design variables are the diffusion coefficient (a, determining how quickly the quantity spreads through the domain) and the reaction coefficient (b, determining how the quantity decays or grows locally). The controller gains k1 and k2 are also optimized. This is the most complex scenario, with all four variables treated as unknowns to be chosen simultaneously.

Case 2: Optimization of only the diffusion coefficient. The reaction coefficient is held fixed while the diffusion coefficient and both control gains are optimized.

Case 3: Fixed physical parameters, control-only optimization. The design variables are held constant, and only the controller gains are optimized. This corresponds to the traditional sequential design approach and serves as a baseline for comparison.

Case 4: Optimization of both physical parameters with zero cost on control effort. This scenario examines what happens when the designer prioritizes performance over energy efficiency, effectively setting the weight on control effort to zero.

For each case, the researchers track the convergence of the gradient descent algorithm and report the final values of the design parameters and controller gains, along with the resulting cost function values. The numerical experiments demonstrate that the algorithm successfully converges to stable solutions that satisfy all the sufficient stability conditions, and that co-design solutions achieve lower cost values than sequential design approaches.

The figures in the paper illustrate the evolution of the system state under different scenarios.

shows the uncontrolled response of the system—the temperature or concentration profile diverging from its target state over time. This serves as a baseline demonstrating why control is necessary.

shows the system behavior at initialization of the CCD algorithm, before optimization has occurred.

shows the optimal CCD solution, where the system has been designed to work in concert with its controller to achieve the best possible performance.

The numerical results indicate that when physical design parameters are included in the optimization, the resulting cost is lower than when only control gains are optimized. This is the fundamental promise of control co-design: by treating the system and its controller as jointly optimizable, you can find designs that neither the best sequential approach nor the best controller-for-a-fixed-system could achieve.

The researchers note that the sufficient stability conditions they derive are conservative—meaning they identify a subset of the true feasible region. This is a limitation inherent to Lyapunov-based stability analysis: the conditions are sufficient but not necessary. As a result, some optimal designs might be excluded by the constraints. However, the conservative nature of the conditions is offset by their simplicity and computational tractability.

The spatial discretization is another source of approximation. As the number of discretization points increases, the discretized system more closely approximates the continuous PDE, but computational cost grows accordingly. The researchers use a finite difference scheme with ghost nodes to handle the Neumann boundary conditions, where the spatial derivative at the boundary is specified (in this case, proportional to the state at the boundary). The convergence of the method as the discretization becomes finer is left as a theoretical question for future work.

Why This Changes Things

The gap between theory and practice in control engineering is often wider than it appears. Mathematical results that seem powerful in the abstract can fail to translate to real-world impact if they're too complex to implement or too brittle to handle real-world uncertainties. What Yadav and Chanekar have done is not just prove a theorem—it's provide a computable path from mathematical elegance to engineering utility.

Consider the practical implications. In traditional sequential design, the hardware team builds a system to specifications, then hands it off to the controls team, who must make do with whatever they've been given. If the system turns out to have properties that make it hard to control—or if a slight change in the material properties would make control dramatically easier—the sequential approach can't exploit this insight. The design team doesn't know what the controls team needs; the controls team can't influence what the design team builds.

Control co-design breaks down this barrier. By formulating the problem so that both the physical parameters and the control gains appear as optimization variables, the method inherently finds designs that are "controllable by construction." A heat exchanger designed this way wouldn't just be one that can be controlled—it would be one that was optimized together with its controller from the start.

The focus on boundary control is particularly important for practical implementation. Many real-world systems can only be actuated at their boundaries—you can heat one end of a rod, but you can't inject heat directly into the middle. Boundary control is simpler to implement than distributed actuation, and it requires fewer sensors and actuators. By developing conditions for stability under boundary feedback, Yadav and Chanekar keep their method within the realm of practical engineering.

The reaction-diffusion context of the PDE they study covers a wide range of applications. Thermal regulation systems—crucial for everything from spacecraft to data centers—fit squarely into this framework. Chemical process control, where concentration gradients drive reactions, is another natural application. The spread of pollutants in groundwater, the propagation of stress waves in materials, and the dynamics of populations moving through habitats all fall within the class of systems this theory addresses.

There's also a subtler point about optimization methodology. The gradient-based approach the researchers develop is inherently local—it finds a locally optimal solution, not necessarily the global optimum. This is true of most practical optimization algorithms. But the structure of the problem, combined with the stability constraints, may make local optima good enough. And unlike methods that require global exploration of a nonconvex landscape, gradient descent is computationally efficient and scales reasonably to higher-dimensional problems.

The numerical example demonstrates convergence across different scenarios, showing that the algorithm doesn't get stuck in infeasible regions or fail to find stable designs. This is important for practical adoption: engineers need to trust that the method will work reliably, not just theoretically but in implementation.

What's Next

The paper opens several threads for future research. The most immediate is relaxation of the sufficient conditions. The Lyapunov-based stability analysis is conservative by construction. Future work might derive less restrictive conditions—either through different Lyapunov functionals or through direct analysis of the discretized system—expanding the set of feasible designs and potentially finding better solutions.

The convergence properties of the gradient descent algorithm also warrant deeper investigation. The paper establishes that the algorithm converges in numerical experiments, but theoretical guarantees about convergence rate, robustness to initialization, and behavior on higher-dimensional discretizations would strengthen the practical foundations. This connects to broader questions in non-convex optimization, where gradient methods can sometimes converge to saddle points or local minima that aren't globally optimal.

The extension to more complex PDEs is a natural direction. The paper focuses on a one-dimensional parabolic PDE with constant coefficients. Real-world systems often involve higher dimensions (three-dimensional spatial domains), variable coefficients, nonlinear dynamics, or coupled systems of PDEs. Each generalization introduces mathematical challenges that may require fundamentally new approaches.

Uncertainty is another gap. The formulation assumes perfect knowledge of the system parameters and complete controllability of the boundaries. Real systems face disturbances, measurement noise, parameter uncertainty, and model errors. Robust control co-design—where the design must perform well not just for one specific system but across a range of possible variations—would significantly increase practical utility.

Experimental validation remains crucial. The numerical examples demonstrate the method on a synthetic problem where the true behavior can be computed and compared. Testing on physical systems—implementing the boundary control laws and comparing against sequential design approaches—would provide the kind of evidence that drives adoption in industry.

The relationship to existing industrial practice also deserves examination. Control co-design has gained traction in certain domains, particularly aerospace and automotive engineering, where the systems are complex enough that the performance gains justify the methodological shift. The methods developed here might find the most immediate application in process industries, where thermal and chemical systems are common and where the economic incentives for optimization are strong.

Perhaps most fundamentally, the paper demonstrates that the computational intractability of infinite-dimensional co-design can be overcome. By combining rigorous stability analysis with smart discretization and gradient-based optimization, Yadav and Chanekar have shown a path forward that respects both mathematical precision and computational reality. Whether this particular formulation will be the one that finds widespread application, or whether it will inspire variations that do, the contribution to the toolkit of control co-design is substantial.

The broader lesson is about integration. Engineering has always involved trade-offs between analysis and design, between models and reality, between optimal and implementable. Control co-design represents an attempt to collapse these dichotomies—to find designs that are optimal not in isolation but in the context of how they will actually be controlled. For the class of systems studied in this paper, that integration is now on a firmer mathematical footing.